![\"Think](\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/04\/Screenshot-2026-04-14-at-12.30.40-PM.png\")
<\/a>\n\n\n\n
What are equivalent fractions?<\/strong><\/h2>\n\n\n\nEquivalent fractions are fractions that have the same value when simplified, even though their numerators and denominators are different numbers. Two fractions are equivalent if dividing the numerator by the denominator gives the same result for both.<\/p>\n\n\n
\n![\"equivalent](\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/04\/Screenshot-2026-04-28-at-11.15.07-AM.png\")
<\/figure>\n<\/div>\n\n\nThe simplest example is 1\/2 and 2\/4. Dividing 1 by 2 gives 0.5. Dividing 2 by 4 also gives 0.5. The fractions look different but represent exactly the same quantity.<\/p>\n\n\n\n
A fraction and all its equivalent fractions form a family. Every member of the family can be obtained from any other member by multiplying or dividing both the numerator and denominator by the same non-zero number. <\/p>\n\n\n\n
The best way to understand equivalent fractions is by trying out problems with them yourself. This article contains an equivalent fractions worksheet for you to get your own experience with them. <\/p>\n\n\n\n
The fundamental rule of equivalent fractions<\/strong><\/h3>\n\n\n\nMultiplying or dividing both the numerator and denominator of a fraction by the same non-zero number produces an equivalent fraction.<\/p>\n\n\n\n
If a\/b is a fraction and k is any non-zero number:<\/p>\n\n\n\n
(a x k) \/ (b x k) = a\/b<\/strong><\/p>\n\n\n\nThis rule works in both directions:<\/p>\n\n\n\n
\n- Multiplying both by k scales the fraction up to a larger equivalent fraction<\/li>\n\n\n\n
- Dividing both by k scales it down to a smaller equivalent fraction \u2014 this is called simplifying<\/li>\n<\/ul>\n\n\n\n
Why equivalent fractions matter in AMC 8<\/strong><\/h3>\n\n\n\nEquivalent fractions appear in AMC 8 problems in several forms. Comparing fractions requires finding a common denominator, which means converting fractions into equivalent forms with the same denominator. Probability problems often produce fractions that need simplifying to match one of the answer choices. Ratio problems require recognising when two ratios represent the same relationship. Students who can quickly generate and identify equivalent fractions by practising with an equivalent fractions worksheet solve these problems significantly faster than students who work from first principles each time.<\/p>\n\n\n\n
\n\n\n\nTwo methods for finding equivalent fractions<\/strong><\/h2>\n\n\n\nMethod one \u2014 multiply numerator and denominator<\/strong><\/h3>\n\n\n\nTo scale a fraction up, multiply both the numerator and denominator by the same number.<\/p>\n\n\n\n
Example: Find three fractions equivalent to 3\/5.<\/p>\n\n\n\n
Multiply by 2: (3 x 2)\/(5 x 2) = 6\/10 Multiply by 3: (3 x 3)\/(5 x 3) = 9\/15 Multiply by 4: (3 x 4)\/(5 x 4) = 12\/20<\/p>\n\n\n\n
All of 6\/10, 9\/15, and 12\/20 are equivalent to 3\/5.<\/p>\n\n\n\n
Method two \u2014 divide numerator and denominator (simplifying)<\/strong><\/h3>\n\n\n\nTo scale a fraction down, divide both the numerator and denominator by their greatest common factor.<\/p>\n\n\n\n
Example: Simplify 18\/24 to its lowest terms.<\/p>\n\n\n\n
GCF of 18 and 24 = 6 (18 \/ 6) \/ (24 \/ 6) = 3\/4<\/p>\n\n\n\n
So 18\/24 is equivalent to 3\/4 in its simplest form.<\/p>\n\n\n\n
For more on GCF, including questions and solutions, read: What is the GCF? How to Find the Greatest Common Factor With Examples<\/a>.<\/p>\n\n\n\nHow to check if two fractions are equivalent<\/strong><\/h3>\n\n\n\nUse cross multiplication. If a\/b and c\/d are equivalent then a x d = b x c.<\/p>\n\n\n
\n![\"how](\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/04\/Screenshot-2026-04-28-at-11.18.22-AM-1.png\")
<\/figure>\n<\/div>\n\n\nExample: Check whether 4\/6 and 10\/15 are equivalent.<\/p>\n\n\n\n
4 x 15 = 60 <\/p>\n\n\n\n
6 x 10 = 60<\/p>\n\n\n\n
The cross products are equal, so the fractions are equivalent.<\/p>\n\n\n\n
Example: Check whether 3\/7 and 5\/12 are equivalent.<\/p>\n\n\n\n
3 x 12 = 36 <\/p>\n\n\n\n
7 x 5 = 35<\/p>\n\n\n\n
The cross products are not equal, so the fractions are not equivalent.<\/p>\n\n\n\n
\n\n\n\nFractional equivalent \u2014 finding a missing numerator or denominator<\/strong><\/h2>\n\n\n\nMany AMC 8 problems and this equivalent fractions worksheet ask you to find the missing number that makes two fractions equivalent. This is called finding the fractional equivalent and uses the same cross multiplication principle.<\/p>\n\n\n\n
Finding a missing numerator<\/strong><\/h3>\n\n\n\nExample: Find \ud835\udc65 such that \ud835\udc65\/20 = 3\/4.<\/p>\n\n\n\n
Cross multiply: \ud835\udc65 x 4 = 20 x 3 <\/p>\n\n\n\n
4\ud835\udc65 = 60 <\/p>\n\n\n\n
\ud835\udc65 = 15<\/p>\n\n\n\n
So 15\/20 is the fractional equivalent of 3\/4 with denominator 20.<\/p>\n\n\n\n
Alternative method:<\/strong> The denominator went from 4 to 20 \u2014 it was multiplied by 5. So multiply the numerator by 5 as well: 3 x 5 = 15.<\/p>\n\n\n\nFinding a missing denominator<\/strong><\/h3>\n\n\n\nExample: Find y such that 7\/y = 21\/45.<\/p>\n\n\n\n
Cross multiply: 7 x 45 = 21 x y <\/p>\n\n\n\n
315 = 21y <\/p>\n\n\n\n
y = 15<\/p>\n\n\n\n
So 7\/15 is the fractional equivalent of 21\/45.<\/p>\n\n\n\n
Alternative method:<\/strong> The numerator went from 7 to 21 \u2014 it was multiplied by 3. So multiply the denominator by 3 as well: 5 x 3 = 15.<\/p>\n\n\n\nWhen to use cross multiplication vs the scaling method<\/strong><\/h3>\n\n\n\nFor simple fractions where the scaling factor is obvious, the scaling method is faster. If 2\/3 = \ud835\udc65\/12, it is immediately clear that the denominator was multiplied by 4, so the numerator is 8.<\/p>\n\n\n\n
For less obvious fractions, cross multiplication is more reliable. Always use cross multiplication when the scaling factor is not immediately apparent or when verifying whether two fractions are equivalent.<\/p>\n\n\n\n
\n\n\n\nEquivalent fractions worksheet \u2014 Level 1 problems<\/strong><\/h2>\n\n\n\nThese problems test direct application of the equivalent fractions rule. They are equivalent in difficulty to questions 1 to 8 on the AMC 8.<\/p>\n\n\n\n
\n\n\n\nProblem 1<\/strong><\/p>\n\n\n\nFind four fractions equivalent to 2\/3.<\/p>\n\n\n\n
Solution:<\/strong> Multiply numerator and denominator by 2, 3, 4, and 5: 4\/6, 6\/9, 8\/12, 10\/15<\/p>\n\n\n\nAnswer: 4\/6, 6\/9, 8\/12, 10\/15<\/strong> (any four correct answers accepted)<\/p>\n\n\n\n
\n\n\n\nProblem 2<\/strong><\/p>\n\n\n\nSimplify 36\/48 to its lowest terms.<\/p>\n\n\n\n
Solution:<\/strong> Find the GCF of 36 and 48. 36 = 2\u00b2 x 3\u00b2 <\/p>\n\n\n\n48 = 2\u2074 x 3 <\/p>\n\n\n\n
GCF = 2\u00b2 x 3 = 12<\/p>\n\n\n\n
36\/12 = 3 <\/p>\n\n\n\n
48\/12 = 4<\/p>\n\n\n\n
Answer: 3\/4<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 3<\/strong><\/p>\n\n\n\nFind \ud835\udc65 such that \ud835\udc65\/35 = 4\/7.<\/p>\n\n\n\n
Solution:<\/strong> Denominator scaled from 7 to 35 \u2014 multiplied by 5. Numerator: 4 x 5 = 20.<\/p>\n\n\n\nAnswer: \ud835\udc65 = 20<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 4<\/strong><\/p>\n\n\n\nAre 15\/25 and 9\/15 equivalent fractions?<\/p>\n\n\n\n
Solution:<\/strong> Cross multiply: 15 x 15 = 225 and 25 x 9 = 225. Equal cross products \u2014 yes, they are equivalent.<\/p>\n\n\n\nAlternatively, simplify both: 15\/25 = 3\/5 (divide by 5) <\/p>\n\n\n\n
9\/15 = 3\/5 (divide by 3) <\/p>\n\n\n\n
Both simplify to 3\/5 \u2014 equivalent.<\/p>\n\n\n\n
Answer: Yes<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 5<\/strong><\/p>\n\n\n\nWrite 5\/8 as an equivalent fraction with denominator 56.<\/p>\n\n\n\n
Solution:<\/strong> 56 \/ 8 = 7. Multiply numerator by 7: 5 x 7 = 35.<\/p>\n\n\n\nAnswer: 35\/56<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 6<\/strong><\/p>\n\n\n\nWhich of the following is not equivalent to 2\/5? (A) 4\/10 <\/p>\n\n\n\n
(B) 6\/15 <\/p>\n\n\n\n
(C) 8\/25 <\/p>\n\n\n\n
(D) 10\/25 <\/p>\n\n\n\n
(E) 14\/35<\/p>\n\n\n\n
Solution:<\/strong> Check each by simplifying or cross-multiplying: <\/p>\n\n\n\n4\/10 = 2\/5 \u2713 <\/p>\n\n\n\n
6\/15 = 2\/5 \u2713 <\/p>\n\n\n\n
8\/25: cross multiply 8 x 5 = 40, 25 x 2 = 50. <\/p>\n\n\n\n
Not equal \u2717 <\/p>\n\n\n\n
No need to check further.<\/p>\n\n\n\n
Answer: (C) 8\/25<\/strong><\/p>\n\n\n\nKey insight:<\/strong> This is the style of an AMC 8 multiple-choice problem involving equivalent fractions. The fastest approach is to simplify each option and check whether it equals 2\/5, or cross multiply. Checking in order from A ensures you do not skip the correct answer.<\/p>\n\n\n\n
\n\n\n\nEquivalent fractions worksheet \u2014 Level 2 problems<\/strong><\/h2>\n\n\n\nThese problems require applying equivalent fractions in a broader context \u2014 comparing, ordering, and using them in calculations. They are equivalent in difficulty to questions 8 to 16 on the AMC 8.<\/p>\n\n\n\n
\n\n\n\nProblem 7 \u2014 comparing fractions<\/strong><\/p>\n\n\n\nWhich is greater, 7\/12 or 5\/9?<\/p>\n\n\n\n
Solution:<\/strong> Find equivalent fractions with a common denominator. LCM of 12 and 9 = 36.<\/p>\n\n\n\n7\/12 = 21\/36 (multiply by 3) <\/p>\n\n\n\n
5\/9 = 20\/36 (multiply by 4)<\/p>\n\n\n\n
21\/36 > 20\/36, so 7\/12 > 5\/9.<\/p>\n\n\n\n
Answer: 7\/12 is greater<\/strong><\/p>\n\n\n\nKey insight:<\/strong> Comparing fractions always requires a common denominator. Find the LCM of the two denominators, convert each fraction to an equivalent form with that denominator, then compare the numerators directly.<\/p>\n\n\n\n
\n\n\n\nProblem 8 \u2014 ordering fractions<\/strong><\/p>\n\n\n\nArrange in ascending order: 3\/4, 2\/3, 5\/6, 7\/12.<\/p>\n\n\n\n
Solution:<\/strong> Common denominator = 12.<\/p>\n\n\n\n3\/4 = 9\/12 <\/p>\n\n\n\n
2\/3 = 8\/12 <\/p>\n\n\n\n
5\/6 = 10\/12 <\/p>\n\n\n\n
7\/12 = 7\/12<\/p>\n\n\n\n
Ascending order: 7\/12, 8\/12, 9\/12, 10\/12 <\/p>\n\n\n\n
Which is: 7\/12, 2\/3, 3\/4, 5\/6.<\/p>\n\n\n\n
Answer: 7\/12, 2\/3, 3\/4, 5\/6<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 9 \u2014 ratio problem<\/strong><\/p>\n\n\n\nA recipe uses flour and sugar in the ratio 3:5. If 24 grams of flour are used, how many grams of sugar are needed?<\/p>\n\n\n\n
Solution:<\/strong> 3\/5 = 24\/\ud835\udc65 <\/p>\n\n\n\nCross multiply: 3\ud835\udc65 = 120 <\/p>\n\n\n\n
\ud835\udc65 = 40<\/p>\n\n\n\n
Answer: 40 grams of sugar<\/strong><\/p>\n\n\n\nKey insight:<\/strong> Ratio problems are equivalent fraction problems in disguise. Setting up the proportion as two equivalent fractions and cross-multiplying is the most reliable method.<\/p>\n\n\n\n
\n\n\n\nProblem 10 \u2014 probability<\/strong><\/p>\n\n\n\nA bag contains 12 red marbles and 8 blue marbles. What fraction of the marbles are red? Express in simplest form.<\/p>\n\n\n\n
Solution:<\/strong> Total marbles = 20. <\/p>\n\n\n\nFraction red = 12\/20. <\/p>\n\n\n\n
Simplify: GCF of 12 and 20 = 4. <\/p>\n\n\n\n
12\/20 = 3\/5.<\/p>\n\n\n\n
Answer: 3\/5<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 11 \u2014 missing value in a proportion<\/strong><\/p>\n\n\n\nIf 3\/\ud835\udc65 = \ud835\udc65\/12, find the positive value of \ud835\udc65. <\/p>\n\n\n\n
Solution:<\/strong> Cross multiply: 3 x 12 =\ud835\udc65 x \ud835\udc65 <\/p>\n\n\n\n36 = \ud835\udc65\u00b2 <\/p>\n\n\n\n
\ud835\udc65 = 6<\/p>\n\n\n\n
Answer: \ud835\udc65 = 6<\/strong><\/p>\n\n\n\nKey insight:<\/strong> This type of problem \u2014 finding the geometric mean \u2014 uses cross multiplication on a proportion where the same variable appears in both fractions. It appears in AMC 8 problems involving similar triangles and proportional reasoning.<\/p>\n\n\n\n
\n\n\n\nProblem 12 \u2014 equivalent fractions with variables<\/strong><\/p>\n\n\n\nIf (2\ud835\udc65)\/(3\ud835\udc65 + 1) = 6\/10, find \ud835\udc65.<\/p>\n\n\n\n
Solution:<\/strong> Cross multiply: 2\ud835\udc65 x 10 = 6 x (3\ud835\udc65 + 1) <\/p>\n\n\n\n20\ud835\udc65 = 18\ud835\udc65+ 6 <\/p>\n\n\n\n
2\ud835\udc65 = 6\ud835\udc65 = 3<\/p>\n\n\n\n
Verification:<\/strong> (2 x 3)\/(3 x 3 + 1) = 6\/10. 6\/10 = 6\/10 \u2713<\/p>\n\n\n\nAnswer: \ud835\udc65 = 3<\/strong><\/p>\n\n\n\n![\"Think](\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/04\/Screenshot-2026-04-14-at-12.30.45-PM.png\")
<\/a>Think Academy students have earned 1,700+ AMC 8 medals since 2021. Find the right course level for your child.<\/figcaption><\/figure>\n\n\n\n
\n\n\n\nEquivalent fractions worksheet \u2014 AMC 8 style problems<\/strong><\/h2>\n\n\n\nThese problems are written in the style of actual AMC 8 past contest questions. They require recognising that equivalent fractions are involved without being told explicitly.<\/p>\n\n\n\n
\n\n\n\nProblem 13<\/strong><\/p>\n\n\n\nWhat fraction of the numbers from 1 to 30 are divisible by 6? Express as a fraction in simplest form.<\/p>\n\n\n\n
Solution:<\/strong> Numbers divisible by 6: 6, 12, 18, 24, 30 \u2014 five numbers. <\/p>\n\n\n\nFraction = 5\/30. <\/p>\n\n\n\n
Simplify: GCF of 5 and 30 = 5. <\/p>\n\n\n\n
5\/30 = 1\/6.<\/p>\n\n\n\n
Answer: 1\/6<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 14<\/strong><\/p>\n\n\n\nA school has 360 students. If 2\/5 of them play sport and 3\/4 of those who play sport also play a musical instrument, how many students play both sport and a musical instrument?<\/p>\n\n\n\n
Solution:<\/strong> Students who play sport = 2\/5 x 360 = 144. <\/p>\n\n\n\nStudents who play both = 3\/4 x 144 = 108.<\/p>\n\n\n\n
Answer: 108<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 15<\/strong><\/p>\n\n\n\nTwo fractions have a sum of 1. One fraction is 3\/7. What is the other fraction in simplest form?<\/p>\n\n\n\n
Solution:<\/strong> Other fraction = 1 – 3\/7 = 7\/7 – 3\/7 = 4\/7.<\/p>\n\n\n\nAnswer: 4\/7<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 16<\/strong><\/p>\n\n\n\nThree fractions are in the ratio 1:2:3, and their sum is 1. What is the largest fraction?<\/p>\n\n\n\n
Solution:<\/strong> Let the fractions be k, 2k, and 3k. k + 2k + 3k = 1 <\/p>\n\n\n\n6k = 1 k = 1\/6<\/p>\n\n\n\n
Largest fraction = 3k = 3\/6 = 1\/2.<\/p>\n\n\n\n
Answer: 1\/2<\/strong><\/p>\n\n\n\nKey insight:<\/strong> When fractions are in a given ratio and their sum is known, introduce a variable k, write each fraction as a multiple of k, and set their sum equal to the known total.<\/p>\n\n\n\n
\n\n\n\nProblem 17<\/strong><\/p>\n\n\n\nIn a class of 40 students, the ratio of boys to girls is 3:5. How many girls are in the class?<\/p>\n\n\n\n
Solution:<\/strong> Total parts = 3 + 5 = 8. <\/p>\n\n\n\nGirls = 5\/8 x 40 = 25.<\/p>\n\n\n\n
Answer: 25<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 18<\/strong><\/p>\n\n\n\nA number is multiplied by 3\/4, and then the result is multiplied by 8\/9. The final result is 2. What was the original number?<\/p>\n\n\n\n
Solution:<\/strong> Let the number be \ud835\udc65. <\/p>\n\n\n\n\ud835\udc65 x 3\/4 x 8\/9 = 2\ud835\udc65 x 24\/36 = 2\ud835\udc65 x 2\/3 = 2\ud835\udc65 = 3<\/p>\n\n\n\n
Verification:<\/strong> 3 x 3\/4 = 9\/4. 9\/4 x 8\/9 = 72\/36 = 2 \u2713<\/p>\n\n\n\nAnswer: 3<\/strong><\/p>\n\n\n\nKey insight:<\/strong> Multiply the fractions together first before solving for \ud835\udc65 \u2014 3\/4 x 8\/9 = 24\/36 = 2\/3. This simplification step using equivalent fractions makes the arithmetic much cleaner.<\/p>\n\n\n\n
\n\n\n\nProblem 19<\/strong><\/p>\n\n\n\nWhich value of n makes the fractions 4\/7, n\/21, and 20\/35 all equivalent?<\/p>\n\n\n\n
Solution:<\/strong> Simplify 20\/35: GCF of 20 and 35 = 5. 20\/35 = 4\/7. So the target fraction is 4\/7.<\/p>\n\n\n\nFor n\/21: 21\/7 = 3. So n = 4 x 3 = 12.<\/p>\n\n\n\n
Verification:<\/strong> 4\/7 = 12\/21 = 20\/35. Cross multiply to check: 4 x 21 = 84, 7 x 12 = 84 \u2713<\/p>\n\n\n\nAnswer: n = 12<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 20 \u2014 hardest<\/strong><\/p>\n\n\n\nThe fraction (2a + 3)\/(3a + 5) equals 5\/7 for some value a. What is the value of 4a?<\/p>\n\n\n\n
Solution:<\/strong> Cross multiply: 7(2a + 3) = 5(3a + 5) <\/p>\n\n\n\n14a + 21 = 15a + 25 <\/p>\n\n\n\n
14a – 15a = 25 – 21 -a = 4 a = -4<\/p>\n\n\n\n
4a = -16<\/p>\n\n\n\n
Verification:<\/strong> (2(-4) + 3)\/(3(-4) + 5) = (-8 + 3)\/(-12 + 5) = -5\/-7 = 5\/7 \u2713<\/p>\n\n\n\nAnswer: 4a = -16<\/strong><\/p>\n\n\n\nKey insight:<\/strong> The question asks for 4a, not a \u2014 re-read the question before writing the final answer. This is a deliberate AMC technique to catch students who solve correctly but answer the wrong thing.<\/p>\n\n\n\n
\n\n\n\nEquivalent fractions reference sheet<\/strong><\/h2>\n\n\n\n
<\/figure>\n<\/div>\n\n\nThe simplest example is 1\/2 and 2\/4. Dividing 1 by 2 gives 0.5. Dividing 2 by 4 also gives 0.5. The fractions look different but represent exactly the same quantity.<\/p>\n\n\n\n
A fraction and all its equivalent fractions form a family. Every member of the family can be obtained from any other member by multiplying or dividing both the numerator and denominator by the same non-zero number. <\/p>\n\n\n\n
The best way to understand equivalent fractions is by trying out problems with them yourself. This article contains an equivalent fractions worksheet for you to get your own experience with them. <\/p>\n\n\n\n
The fundamental rule of equivalent fractions<\/strong><\/h3>\n\n\n\nMultiplying or dividing both the numerator and denominator of a fraction by the same non-zero number produces an equivalent fraction.<\/p>\n\n\n\n
If a\/b is a fraction and k is any non-zero number:<\/p>\n\n\n\n
(a x k) \/ (b x k) = a\/b<\/strong><\/p>\n\n\n\nThis rule works in both directions:<\/p>\n\n\n\n
\n- Multiplying both by k scales the fraction up to a larger equivalent fraction<\/li>\n\n\n\n
- Dividing both by k scales it down to a smaller equivalent fraction \u2014 this is called simplifying<\/li>\n<\/ul>\n\n\n\n
Why equivalent fractions matter in AMC 8<\/strong><\/h3>\n\n\n\nEquivalent fractions appear in AMC 8 problems in several forms. Comparing fractions requires finding a common denominator, which means converting fractions into equivalent forms with the same denominator. Probability problems often produce fractions that need simplifying to match one of the answer choices. Ratio problems require recognising when two ratios represent the same relationship. Students who can quickly generate and identify equivalent fractions by practising with an equivalent fractions worksheet solve these problems significantly faster than students who work from first principles each time.<\/p>\n\n\n\n
\n\n\n\nTwo methods for finding equivalent fractions<\/strong><\/h2>\n\n\n\nMethod one \u2014 multiply numerator and denominator<\/strong><\/h3>\n\n\n\nTo scale a fraction up, multiply both the numerator and denominator by the same number.<\/p>\n\n\n\n
Example: Find three fractions equivalent to 3\/5.<\/p>\n\n\n\n
Multiply by 2: (3 x 2)\/(5 x 2) = 6\/10 Multiply by 3: (3 x 3)\/(5 x 3) = 9\/15 Multiply by 4: (3 x 4)\/(5 x 4) = 12\/20<\/p>\n\n\n\n
All of 6\/10, 9\/15, and 12\/20 are equivalent to 3\/5.<\/p>\n\n\n\n
Method two \u2014 divide numerator and denominator (simplifying)<\/strong><\/h3>\n\n\n\nTo scale a fraction down, divide both the numerator and denominator by their greatest common factor.<\/p>\n\n\n\n
Example: Simplify 18\/24 to its lowest terms.<\/p>\n\n\n\n
GCF of 18 and 24 = 6 (18 \/ 6) \/ (24 \/ 6) = 3\/4<\/p>\n\n\n\n
So 18\/24 is equivalent to 3\/4 in its simplest form.<\/p>\n\n\n\n
For more on GCF, including questions and solutions, read: What is the GCF? How to Find the Greatest Common Factor With Examples<\/a>.<\/p>\n\n\n\nHow to check if two fractions are equivalent<\/strong><\/h3>\n\n\n\nUse cross multiplication. If a\/b and c\/d are equivalent then a x d = b x c.<\/p>\n\n\n
\n<\/figure>\n<\/div>\n\n\nExample: Check whether 4\/6 and 10\/15 are equivalent.<\/p>\n\n\n\n
4 x 15 = 60 <\/p>\n\n\n\n
6 x 10 = 60<\/p>\n\n\n\n
The cross products are equal, so the fractions are equivalent.<\/p>\n\n\n\n
Example: Check whether 3\/7 and 5\/12 are equivalent.<\/p>\n\n\n\n
3 x 12 = 36 <\/p>\n\n\n\n
7 x 5 = 35<\/p>\n\n\n\n
The cross products are not equal, so the fractions are not equivalent.<\/p>\n\n\n\n
\n\n\n\nFractional equivalent \u2014 finding a missing numerator or denominator<\/strong><\/h2>\n\n\n\nMany AMC 8 problems and this equivalent fractions worksheet ask you to find the missing number that makes two fractions equivalent. This is called finding the fractional equivalent and uses the same cross multiplication principle.<\/p>\n\n\n\n
Finding a missing numerator<\/strong><\/h3>\n\n\n\nExample: Find \ud835\udc65 such that \ud835\udc65\/20 = 3\/4.<\/p>\n\n\n\n
Cross multiply: \ud835\udc65 x 4 = 20 x 3 <\/p>\n\n\n\n
4\ud835\udc65 = 60 <\/p>\n\n\n\n
\ud835\udc65 = 15<\/p>\n\n\n\n
So 15\/20 is the fractional equivalent of 3\/4 with denominator 20.<\/p>\n\n\n\n
Alternative method:<\/strong> The denominator went from 4 to 20 \u2014 it was multiplied by 5. So multiply the numerator by 5 as well: 3 x 5 = 15.<\/p>\n\n\n\nFinding a missing denominator<\/strong><\/h3>\n\n\n\nExample: Find y such that 7\/y = 21\/45.<\/p>\n\n\n\n
Cross multiply: 7 x 45 = 21 x y <\/p>\n\n\n\n
315 = 21y <\/p>\n\n\n\n
y = 15<\/p>\n\n\n\n
So 7\/15 is the fractional equivalent of 21\/45.<\/p>\n\n\n\n
Alternative method:<\/strong> The numerator went from 7 to 21 \u2014 it was multiplied by 3. So multiply the denominator by 3 as well: 5 x 3 = 15.<\/p>\n\n\n\nWhen to use cross multiplication vs the scaling method<\/strong><\/h3>\n\n\n\nFor simple fractions where the scaling factor is obvious, the scaling method is faster. If 2\/3 = \ud835\udc65\/12, it is immediately clear that the denominator was multiplied by 4, so the numerator is 8.<\/p>\n\n\n\n
For less obvious fractions, cross multiplication is more reliable. Always use cross multiplication when the scaling factor is not immediately apparent or when verifying whether two fractions are equivalent.<\/p>\n\n\n\n
\n\n\n\nEquivalent fractions worksheet \u2014 Level 1 problems<\/strong><\/h2>\n\n\n\nThese problems test direct application of the equivalent fractions rule. They are equivalent in difficulty to questions 1 to 8 on the AMC 8.<\/p>\n\n\n\n
\n\n\n\nProblem 1<\/strong><\/p>\n\n\n\nFind four fractions equivalent to 2\/3.<\/p>\n\n\n\n
Solution:<\/strong> Multiply numerator and denominator by 2, 3, 4, and 5: 4\/6, 6\/9, 8\/12, 10\/15<\/p>\n\n\n\nAnswer: 4\/6, 6\/9, 8\/12, 10\/15<\/strong> (any four correct answers accepted)<\/p>\n\n\n\n
\n\n\n\nProblem 2<\/strong><\/p>\n\n\n\nSimplify 36\/48 to its lowest terms.<\/p>\n\n\n\n
Solution:<\/strong> Find the GCF of 36 and 48. 36 = 2\u00b2 x 3\u00b2 <\/p>\n\n\n\n48 = 2\u2074 x 3 <\/p>\n\n\n\n
GCF = 2\u00b2 x 3 = 12<\/p>\n\n\n\n
36\/12 = 3 <\/p>\n\n\n\n
48\/12 = 4<\/p>\n\n\n\n
Answer: 3\/4<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 3<\/strong><\/p>\n\n\n\nFind \ud835\udc65 such that \ud835\udc65\/35 = 4\/7.<\/p>\n\n\n\n
Solution:<\/strong> Denominator scaled from 7 to 35 \u2014 multiplied by 5. Numerator: 4 x 5 = 20.<\/p>\n\n\n\nAnswer: \ud835\udc65 = 20<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 4<\/strong><\/p>\n\n\n\nAre 15\/25 and 9\/15 equivalent fractions?<\/p>\n\n\n\n
Solution:<\/strong> Cross multiply: 15 x 15 = 225 and 25 x 9 = 225. Equal cross products \u2014 yes, they are equivalent.<\/p>\n\n\n\nAlternatively, simplify both: 15\/25 = 3\/5 (divide by 5) <\/p>\n\n\n\n
9\/15 = 3\/5 (divide by 3) <\/p>\n\n\n\n
Both simplify to 3\/5 \u2014 equivalent.<\/p>\n\n\n\n
Answer: Yes<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 5<\/strong><\/p>\n\n\n\nWrite 5\/8 as an equivalent fraction with denominator 56.<\/p>\n\n\n\n
Solution:<\/strong> 56 \/ 8 = 7. Multiply numerator by 7: 5 x 7 = 35.<\/p>\n\n\n\nAnswer: 35\/56<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 6<\/strong><\/p>\n\n\n\nWhich of the following is not equivalent to 2\/5? (A) 4\/10 <\/p>\n\n\n\n
(B) 6\/15 <\/p>\n\n\n\n
(C) 8\/25 <\/p>\n\n\n\n
(D) 10\/25 <\/p>\n\n\n\n
(E) 14\/35<\/p>\n\n\n\n
Solution:<\/strong> Check each by simplifying or cross-multiplying: <\/p>\n\n\n\n4\/10 = 2\/5 \u2713 <\/p>\n\n\n\n
6\/15 = 2\/5 \u2713 <\/p>\n\n\n\n
8\/25: cross multiply 8 x 5 = 40, 25 x 2 = 50. <\/p>\n\n\n\n
Not equal \u2717 <\/p>\n\n\n\n
No need to check further.<\/p>\n\n\n\n
Answer: (C) 8\/25<\/strong><\/p>\n\n\n\nKey insight:<\/strong> This is the style of an AMC 8 multiple-choice problem involving equivalent fractions. The fastest approach is to simplify each option and check whether it equals 2\/5, or cross multiply. Checking in order from A ensures you do not skip the correct answer.<\/p>\n\n\n\n
\n\n\n\nEquivalent fractions worksheet \u2014 Level 2 problems<\/strong><\/h2>\n\n\n\nThese problems require applying equivalent fractions in a broader context \u2014 comparing, ordering, and using them in calculations. They are equivalent in difficulty to questions 8 to 16 on the AMC 8.<\/p>\n\n\n\n
\n\n\n\nProblem 7 \u2014 comparing fractions<\/strong><\/p>\n\n\n\nWhich is greater, 7\/12 or 5\/9?<\/p>\n\n\n\n
Solution:<\/strong> Find equivalent fractions with a common denominator. LCM of 12 and 9 = 36.<\/p>\n\n\n\n7\/12 = 21\/36 (multiply by 3) <\/p>\n\n\n\n
5\/9 = 20\/36 (multiply by 4)<\/p>\n\n\n\n
21\/36 > 20\/36, so 7\/12 > 5\/9.<\/p>\n\n\n\n
Answer: 7\/12 is greater<\/strong><\/p>\n\n\n\nKey insight:<\/strong> Comparing fractions always requires a common denominator. Find the LCM of the two denominators, convert each fraction to an equivalent form with that denominator, then compare the numerators directly.<\/p>\n\n\n\n
\n\n\n\nProblem 8 \u2014 ordering fractions<\/strong><\/p>\n\n\n\nArrange in ascending order: 3\/4, 2\/3, 5\/6, 7\/12.<\/p>\n\n\n\n
Solution:<\/strong> Common denominator = 12.<\/p>\n\n\n\n3\/4 = 9\/12 <\/p>\n\n\n\n
2\/3 = 8\/12 <\/p>\n\n\n\n
5\/6 = 10\/12 <\/p>\n\n\n\n
7\/12 = 7\/12<\/p>\n\n\n\n
Ascending order: 7\/12, 8\/12, 9\/12, 10\/12 <\/p>\n\n\n\n
Which is: 7\/12, 2\/3, 3\/4, 5\/6.<\/p>\n\n\n\n
Answer: 7\/12, 2\/3, 3\/4, 5\/6<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 9 \u2014 ratio problem<\/strong><\/p>\n\n\n\nA recipe uses flour and sugar in the ratio 3:5. If 24 grams of flour are used, how many grams of sugar are needed?<\/p>\n\n\n\n
Solution:<\/strong> 3\/5 = 24\/\ud835\udc65 <\/p>\n\n\n\nCross multiply: 3\ud835\udc65 = 120 <\/p>\n\n\n\n
\ud835\udc65 = 40<\/p>\n\n\n\n
Answer: 40 grams of sugar<\/strong><\/p>\n\n\n\nKey insight:<\/strong> Ratio problems are equivalent fraction problems in disguise. Setting up the proportion as two equivalent fractions and cross-multiplying is the most reliable method.<\/p>\n\n\n\n
\n\n\n\nProblem 10 \u2014 probability<\/strong><\/p>\n\n\n\nA bag contains 12 red marbles and 8 blue marbles. What fraction of the marbles are red? Express in simplest form.<\/p>\n\n\n\n
Solution:<\/strong> Total marbles = 20. <\/p>\n\n\n\nFraction red = 12\/20. <\/p>\n\n\n\n
Simplify: GCF of 12 and 20 = 4. <\/p>\n\n\n\n
12\/20 = 3\/5.<\/p>\n\n\n\n
Answer: 3\/5<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 11 \u2014 missing value in a proportion<\/strong><\/p>\n\n\n\nIf 3\/\ud835\udc65 = \ud835\udc65\/12, find the positive value of \ud835\udc65. <\/p>\n\n\n\n
Solution:<\/strong> Cross multiply: 3 x 12 =\ud835\udc65 x \ud835\udc65 <\/p>\n\n\n\n36 = \ud835\udc65\u00b2 <\/p>\n\n\n\n
\ud835\udc65 = 6<\/p>\n\n\n\n
Answer: \ud835\udc65 = 6<\/strong><\/p>\n\n\n\nKey insight:<\/strong> This type of problem \u2014 finding the geometric mean \u2014 uses cross multiplication on a proportion where the same variable appears in both fractions. It appears in AMC 8 problems involving similar triangles and proportional reasoning.<\/p>\n\n\n\n
\n\n\n\nProblem 12 \u2014 equivalent fractions with variables<\/strong><\/p>\n\n\n\nIf (2\ud835\udc65)\/(3\ud835\udc65 + 1) = 6\/10, find \ud835\udc65.<\/p>\n\n\n\n
Solution:<\/strong> Cross multiply: 2\ud835\udc65 x 10 = 6 x (3\ud835\udc65 + 1) <\/p>\n\n\n\n20\ud835\udc65 = 18\ud835\udc65+ 6 <\/p>\n\n\n\n
2\ud835\udc65 = 6\ud835\udc65 = 3<\/p>\n\n\n\n
Verification:<\/strong> (2 x 3)\/(3 x 3 + 1) = 6\/10. 6\/10 = 6\/10 \u2713<\/p>\n\n\n\nAnswer: \ud835\udc65 = 3<\/strong><\/p>\n\n\n\n<\/a>Think Academy students have earned 1,700+ AMC 8 medals since 2021. Find the right course level for your child.<\/figcaption><\/figure>\n\n\n\n
\n\n\n\nEquivalent fractions worksheet \u2014 AMC 8 style problems<\/strong><\/h2>\n\n\n\nThese problems are written in the style of actual AMC 8 past contest questions. They require recognising that equivalent fractions are involved without being told explicitly.<\/p>\n\n\n\n
\n\n\n\nProblem 13<\/strong><\/p>\n\n\n\nWhat fraction of the numbers from 1 to 30 are divisible by 6? Express as a fraction in simplest form.<\/p>\n\n\n\n
Solution:<\/strong> Numbers divisible by 6: 6, 12, 18, 24, 30 \u2014 five numbers. <\/p>\n\n\n\nFraction = 5\/30. <\/p>\n\n\n\n
Simplify: GCF of 5 and 30 = 5. <\/p>\n\n\n\n
5\/30 = 1\/6.<\/p>\n\n\n\n
Answer: 1\/6<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 14<\/strong><\/p>\n\n\n\nA school has 360 students. If 2\/5 of them play sport and 3\/4 of those who play sport also play a musical instrument, how many students play both sport and a musical instrument?<\/p>\n\n\n\n
Solution:<\/strong> Students who play sport = 2\/5 x 360 = 144. <\/p>\n\n\n\nStudents who play both = 3\/4 x 144 = 108.<\/p>\n\n\n\n
Answer: 108<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 15<\/strong><\/p>\n\n\n\nTwo fractions have a sum of 1. One fraction is 3\/7. What is the other fraction in simplest form?<\/p>\n\n\n\n
Solution:<\/strong> Other fraction = 1 – 3\/7 = 7\/7 – 3\/7 = 4\/7.<\/p>\n\n\n\nAnswer: 4\/7<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 16<\/strong><\/p>\n\n\n\nThree fractions are in the ratio 1:2:3, and their sum is 1. What is the largest fraction?<\/p>\n\n\n\n
Solution:<\/strong> Let the fractions be k, 2k, and 3k. k + 2k + 3k = 1 <\/p>\n\n\n\n6k = 1 k = 1\/6<\/p>\n\n\n\n
Largest fraction = 3k = 3\/6 = 1\/2.<\/p>\n\n\n\n
Answer: 1\/2<\/strong><\/p>\n\n\n\nKey insight:<\/strong> When fractions are in a given ratio and their sum is known, introduce a variable k, write each fraction as a multiple of k, and set their sum equal to the known total.<\/p>\n\n\n\n
\n\n\n\nProblem 17<\/strong><\/p>\n\n\n\nIn a class of 40 students, the ratio of boys to girls is 3:5. How many girls are in the class?<\/p>\n\n\n\n
Solution:<\/strong> Total parts = 3 + 5 = 8. <\/p>\n\n\n\nGirls = 5\/8 x 40 = 25.<\/p>\n\n\n\n
Answer: 25<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 18<\/strong><\/p>\n\n\n\nA number is multiplied by 3\/4, and then the result is multiplied by 8\/9. The final result is 2. What was the original number?<\/p>\n\n\n\n
Solution:<\/strong> Let the number be \ud835\udc65. <\/p>\n\n\n\n\ud835\udc65 x 3\/4 x 8\/9 = 2\ud835\udc65 x 24\/36 = 2\ud835\udc65 x 2\/3 = 2\ud835\udc65 = 3<\/p>\n\n\n\n
Verification:<\/strong> 3 x 3\/4 = 9\/4. 9\/4 x 8\/9 = 72\/36 = 2 \u2713<\/p>\n\n\n\nAnswer: 3<\/strong><\/p>\n\n\n\nKey insight:<\/strong> Multiply the fractions together first before solving for \ud835\udc65 \u2014 3\/4 x 8\/9 = 24\/36 = 2\/3. This simplification step using equivalent fractions makes the arithmetic much cleaner.<\/p>\n\n\n\n
\n\n\n\nProblem 19<\/strong><\/p>\n\n\n\nWhich value of n makes the fractions 4\/7, n\/21, and 20\/35 all equivalent?<\/p>\n\n\n\n
Solution:<\/strong> Simplify 20\/35: GCF of 20 and 35 = 5. 20\/35 = 4\/7. So the target fraction is 4\/7.<\/p>\n\n\n\nFor n\/21: 21\/7 = 3. So n = 4 x 3 = 12.<\/p>\n\n\n\n
Verification:<\/strong> 4\/7 = 12\/21 = 20\/35. Cross multiply to check: 4 x 21 = 84, 7 x 12 = 84 \u2713<\/p>\n\n\n\nAnswer: n = 12<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 20 \u2014 hardest<\/strong><\/p>\n\n\n\nThe fraction (2a + 3)\/(3a + 5) equals 5\/7 for some value a. What is the value of 4a?<\/p>\n\n\n\n
Solution:<\/strong> Cross multiply: 7(2a + 3) = 5(3a + 5) <\/p>\n\n\n\n14a + 21 = 15a + 25 <\/p>\n\n\n\n
14a – 15a = 25 – 21 -a = 4 a = -4<\/p>\n\n\n\n
4a = -16<\/p>\n\n\n\n
Verification:<\/strong> (2(-4) + 3)\/(3(-4) + 5) = (-8 + 3)\/(-12 + 5) = -5\/-7 = 5\/7 \u2713<\/p>\n\n\n\nAnswer: 4a = -16<\/strong><\/p>\n\n\n\nKey insight:<\/strong> The question asks for 4a, not a \u2014 re-read the question before writing the final answer. This is a deliberate AMC technique to catch students who solve correctly but answer the wrong thing.<\/p>\n\n\n\n
\n\n\n\nEquivalent fractions reference sheet<\/strong><\/h2>\n\n\n\n
This rule works in both directions:<\/p>\n\n\n\n
- \n
- Multiplying both by k scales the fraction up to a larger equivalent fraction<\/li>\n\n\n\n
- Dividing both by k scales it down to a smaller equivalent fraction \u2014 this is called simplifying<\/li>\n<\/ul>\n\n\n\n
Why equivalent fractions matter in AMC 8<\/strong><\/h3>\n\n\n\n
Equivalent fractions appear in AMC 8 problems in several forms. Comparing fractions requires finding a common denominator, which means converting fractions into equivalent forms with the same denominator. Probability problems often produce fractions that need simplifying to match one of the answer choices. Ratio problems require recognising when two ratios represent the same relationship. Students who can quickly generate and identify equivalent fractions by practising with an equivalent fractions worksheet solve these problems significantly faster than students who work from first principles each time.<\/p>\n\n\n\n
\n\n\n\nTwo methods for finding equivalent fractions<\/strong><\/h2>\n\n\n\n
Method one \u2014 multiply numerator and denominator<\/strong><\/h3>\n\n\n\n
To scale a fraction up, multiply both the numerator and denominator by the same number.<\/p>\n\n\n\n
Example: Find three fractions equivalent to 3\/5.<\/p>\n\n\n\n
Multiply by 2: (3 x 2)\/(5 x 2) = 6\/10 Multiply by 3: (3 x 3)\/(5 x 3) = 9\/15 Multiply by 4: (3 x 4)\/(5 x 4) = 12\/20<\/p>\n\n\n\n
All of 6\/10, 9\/15, and 12\/20 are equivalent to 3\/5.<\/p>\n\n\n\n
Method two \u2014 divide numerator and denominator (simplifying)<\/strong><\/h3>\n\n\n\n
To scale a fraction down, divide both the numerator and denominator by their greatest common factor.<\/p>\n\n\n\n
Example: Simplify 18\/24 to its lowest terms.<\/p>\n\n\n\n
GCF of 18 and 24 = 6 (18 \/ 6) \/ (24 \/ 6) = 3\/4<\/p>\n\n\n\n
So 18\/24 is equivalent to 3\/4 in its simplest form.<\/p>\n\n\n\n
For more on GCF, including questions and solutions, read: What is the GCF? How to Find the Greatest Common Factor With Examples<\/a>.<\/p>\n\n\n\n
How to check if two fractions are equivalent<\/strong><\/h3>\n\n\n\n
Use cross multiplication. If a\/b and c\/d are equivalent then a x d = b x c.<\/p>\n\n\n
\n<\/figure>\n<\/div>\n\n\nExample: Check whether 4\/6 and 10\/15 are equivalent.<\/p>\n\n\n\n
4 x 15 = 60 <\/p>\n\n\n\n
6 x 10 = 60<\/p>\n\n\n\n
The cross products are equal, so the fractions are equivalent.<\/p>\n\n\n\n
Example: Check whether 3\/7 and 5\/12 are equivalent.<\/p>\n\n\n\n
3 x 12 = 36 <\/p>\n\n\n\n
7 x 5 = 35<\/p>\n\n\n\n
The cross products are not equal, so the fractions are not equivalent.<\/p>\n\n\n\n
\n\n\n\nFractional equivalent \u2014 finding a missing numerator or denominator<\/strong><\/h2>\n\n\n\n
Many AMC 8 problems and this equivalent fractions worksheet ask you to find the missing number that makes two fractions equivalent. This is called finding the fractional equivalent and uses the same cross multiplication principle.<\/p>\n\n\n\n
Finding a missing numerator<\/strong><\/h3>\n\n\n\n
Example: Find \ud835\udc65 such that \ud835\udc65\/20 = 3\/4.<\/p>\n\n\n\n
Cross multiply: \ud835\udc65 x 4 = 20 x 3 <\/p>\n\n\n\n
4\ud835\udc65 = 60 <\/p>\n\n\n\n
\ud835\udc65 = 15<\/p>\n\n\n\n
So 15\/20 is the fractional equivalent of 3\/4 with denominator 20.<\/p>\n\n\n\n
Alternative method:<\/strong> The denominator went from 4 to 20 \u2014 it was multiplied by 5. So multiply the numerator by 5 as well: 3 x 5 = 15.<\/p>\n\n\n\n
Finding a missing denominator<\/strong><\/h3>\n\n\n\n
Example: Find y such that 7\/y = 21\/45.<\/p>\n\n\n\n
Cross multiply: 7 x 45 = 21 x y <\/p>\n\n\n\n
315 = 21y <\/p>\n\n\n\n
y = 15<\/p>\n\n\n\n
So 7\/15 is the fractional equivalent of 21\/45.<\/p>\n\n\n\n
Alternative method:<\/strong> The numerator went from 7 to 21 \u2014 it was multiplied by 3. So multiply the denominator by 3 as well: 5 x 3 = 15.<\/p>\n\n\n\n
When to use cross multiplication vs the scaling method<\/strong><\/h3>\n\n\n\n
For simple fractions where the scaling factor is obvious, the scaling method is faster. If 2\/3 = \ud835\udc65\/12, it is immediately clear that the denominator was multiplied by 4, so the numerator is 8.<\/p>\n\n\n\n
For less obvious fractions, cross multiplication is more reliable. Always use cross multiplication when the scaling factor is not immediately apparent or when verifying whether two fractions are equivalent.<\/p>\n\n\n\n
\n\n\n\nEquivalent fractions worksheet \u2014 Level 1 problems<\/strong><\/h2>\n\n\n\n
These problems test direct application of the equivalent fractions rule. They are equivalent in difficulty to questions 1 to 8 on the AMC 8.<\/p>\n\n\n\n
\n\n\n\nProblem 1<\/strong><\/p>\n\n\n\n
Find four fractions equivalent to 2\/3.<\/p>\n\n\n\n
Solution:<\/strong> Multiply numerator and denominator by 2, 3, 4, and 5: 4\/6, 6\/9, 8\/12, 10\/15<\/p>\n\n\n\n
Answer: 4\/6, 6\/9, 8\/12, 10\/15<\/strong> (any four correct answers accepted)<\/p>\n\n\n\n
\n\n\n\nProblem 2<\/strong><\/p>\n\n\n\n
Simplify 36\/48 to its lowest terms.<\/p>\n\n\n\n
Solution:<\/strong> Find the GCF of 36 and 48. 36 = 2\u00b2 x 3\u00b2 <\/p>\n\n\n\n
48 = 2\u2074 x 3 <\/p>\n\n\n\n
GCF = 2\u00b2 x 3 = 12<\/p>\n\n\n\n
36\/12 = 3 <\/p>\n\n\n\n
48\/12 = 4<\/p>\n\n\n\n
Answer: 3\/4<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 3<\/strong><\/p>\n\n\n\n
Find \ud835\udc65 such that \ud835\udc65\/35 = 4\/7.<\/p>\n\n\n\n
Solution:<\/strong> Denominator scaled from 7 to 35 \u2014 multiplied by 5. Numerator: 4 x 5 = 20.<\/p>\n\n\n\n
Answer: \ud835\udc65 = 20<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 4<\/strong><\/p>\n\n\n\n
Are 15\/25 and 9\/15 equivalent fractions?<\/p>\n\n\n\n
Solution:<\/strong> Cross multiply: 15 x 15 = 225 and 25 x 9 = 225. Equal cross products \u2014 yes, they are equivalent.<\/p>\n\n\n\n
Alternatively, simplify both: 15\/25 = 3\/5 (divide by 5) <\/p>\n\n\n\n
9\/15 = 3\/5 (divide by 3) <\/p>\n\n\n\n
Both simplify to 3\/5 \u2014 equivalent.<\/p>\n\n\n\n
Answer: Yes<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 5<\/strong><\/p>\n\n\n\n
Write 5\/8 as an equivalent fraction with denominator 56.<\/p>\n\n\n\n
Solution:<\/strong> 56 \/ 8 = 7. Multiply numerator by 7: 5 x 7 = 35.<\/p>\n\n\n\n
Answer: 35\/56<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 6<\/strong><\/p>\n\n\n\n
Which of the following is not equivalent to 2\/5? (A) 4\/10 <\/p>\n\n\n\n
(B) 6\/15 <\/p>\n\n\n\n
(C) 8\/25 <\/p>\n\n\n\n
(D) 10\/25 <\/p>\n\n\n\n
(E) 14\/35<\/p>\n\n\n\n
Solution:<\/strong> Check each by simplifying or cross-multiplying: <\/p>\n\n\n\n
4\/10 = 2\/5 \u2713 <\/p>\n\n\n\n
6\/15 = 2\/5 \u2713 <\/p>\n\n\n\n
8\/25: cross multiply 8 x 5 = 40, 25 x 2 = 50. <\/p>\n\n\n\n
Not equal \u2717 <\/p>\n\n\n\n
No need to check further.<\/p>\n\n\n\n
Answer: (C) 8\/25<\/strong><\/p>\n\n\n\n
Key insight:<\/strong> This is the style of an AMC 8 multiple-choice problem involving equivalent fractions. The fastest approach is to simplify each option and check whether it equals 2\/5, or cross multiply. Checking in order from A ensures you do not skip the correct answer.<\/p>\n\n\n\n
\n\n\n\nEquivalent fractions worksheet \u2014 Level 2 problems<\/strong><\/h2>\n\n\n\n
These problems require applying equivalent fractions in a broader context \u2014 comparing, ordering, and using them in calculations. They are equivalent in difficulty to questions 8 to 16 on the AMC 8.<\/p>\n\n\n\n
\n\n\n\nProblem 7 \u2014 comparing fractions<\/strong><\/p>\n\n\n\n
Which is greater, 7\/12 or 5\/9?<\/p>\n\n\n\n
Solution:<\/strong> Find equivalent fractions with a common denominator. LCM of 12 and 9 = 36.<\/p>\n\n\n\n
7\/12 = 21\/36 (multiply by 3) <\/p>\n\n\n\n
5\/9 = 20\/36 (multiply by 4)<\/p>\n\n\n\n
21\/36 > 20\/36, so 7\/12 > 5\/9.<\/p>\n\n\n\n
Answer: 7\/12 is greater<\/strong><\/p>\n\n\n\n
Key insight:<\/strong> Comparing fractions always requires a common denominator. Find the LCM of the two denominators, convert each fraction to an equivalent form with that denominator, then compare the numerators directly.<\/p>\n\n\n\n
\n\n\n\nProblem 8 \u2014 ordering fractions<\/strong><\/p>\n\n\n\n
Arrange in ascending order: 3\/4, 2\/3, 5\/6, 7\/12.<\/p>\n\n\n\n
Solution:<\/strong> Common denominator = 12.<\/p>\n\n\n\n
3\/4 = 9\/12 <\/p>\n\n\n\n
2\/3 = 8\/12 <\/p>\n\n\n\n
5\/6 = 10\/12 <\/p>\n\n\n\n
7\/12 = 7\/12<\/p>\n\n\n\n
Ascending order: 7\/12, 8\/12, 9\/12, 10\/12 <\/p>\n\n\n\n
Which is: 7\/12, 2\/3, 3\/4, 5\/6.<\/p>\n\n\n\n
Answer: 7\/12, 2\/3, 3\/4, 5\/6<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 9 \u2014 ratio problem<\/strong><\/p>\n\n\n\n
A recipe uses flour and sugar in the ratio 3:5. If 24 grams of flour are used, how many grams of sugar are needed?<\/p>\n\n\n\n
Solution:<\/strong> 3\/5 = 24\/\ud835\udc65 <\/p>\n\n\n\n
Cross multiply: 3\ud835\udc65 = 120 <\/p>\n\n\n\n
\ud835\udc65 = 40<\/p>\n\n\n\n
Answer: 40 grams of sugar<\/strong><\/p>\n\n\n\n
Key insight:<\/strong> Ratio problems are equivalent fraction problems in disguise. Setting up the proportion as two equivalent fractions and cross-multiplying is the most reliable method.<\/p>\n\n\n\n
\n\n\n\nProblem 10 \u2014 probability<\/strong><\/p>\n\n\n\n
A bag contains 12 red marbles and 8 blue marbles. What fraction of the marbles are red? Express in simplest form.<\/p>\n\n\n\n
Solution:<\/strong> Total marbles = 20. <\/p>\n\n\n\n
Fraction red = 12\/20. <\/p>\n\n\n\n
Simplify: GCF of 12 and 20 = 4. <\/p>\n\n\n\n
12\/20 = 3\/5.<\/p>\n\n\n\n
Answer: 3\/5<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 11 \u2014 missing value in a proportion<\/strong><\/p>\n\n\n\n
If 3\/\ud835\udc65 = \ud835\udc65\/12, find the positive value of \ud835\udc65. <\/p>\n\n\n\n
Solution:<\/strong> Cross multiply: 3 x 12 =\ud835\udc65 x \ud835\udc65 <\/p>\n\n\n\n
36 = \ud835\udc65\u00b2 <\/p>\n\n\n\n
\ud835\udc65 = 6<\/p>\n\n\n\n
Answer: \ud835\udc65 = 6<\/strong><\/p>\n\n\n\n
Key insight:<\/strong> This type of problem \u2014 finding the geometric mean \u2014 uses cross multiplication on a proportion where the same variable appears in both fractions. It appears in AMC 8 problems involving similar triangles and proportional reasoning.<\/p>\n\n\n\n
\n\n\n\nProblem 12 \u2014 equivalent fractions with variables<\/strong><\/p>\n\n\n\n
If (2\ud835\udc65)\/(3\ud835\udc65 + 1) = 6\/10, find \ud835\udc65.<\/p>\n\n\n\n
Solution:<\/strong> Cross multiply: 2\ud835\udc65 x 10 = 6 x (3\ud835\udc65 + 1) <\/p>\n\n\n\n
20\ud835\udc65 = 18\ud835\udc65+ 6 <\/p>\n\n\n\n
2\ud835\udc65 = 6\ud835\udc65 = 3<\/p>\n\n\n\n
Verification:<\/strong> (2 x 3)\/(3 x 3 + 1) = 6\/10. 6\/10 = 6\/10 \u2713<\/p>\n\n\n\n
Answer: \ud835\udc65 = 3<\/strong><\/p>\n\n\n\n
<\/a>Think Academy students have earned 1,700+ AMC 8 medals since 2021. Find the right course level for your child.<\/figcaption><\/figure>\n\n\n\n
\n\n\n\nEquivalent fractions worksheet \u2014 AMC 8 style problems<\/strong><\/h2>\n\n\n\n
These problems are written in the style of actual AMC 8 past contest questions. They require recognising that equivalent fractions are involved without being told explicitly.<\/p>\n\n\n\n
\n\n\n\nProblem 13<\/strong><\/p>\n\n\n\n
What fraction of the numbers from 1 to 30 are divisible by 6? Express as a fraction in simplest form.<\/p>\n\n\n\n
Solution:<\/strong> Numbers divisible by 6: 6, 12, 18, 24, 30 \u2014 five numbers. <\/p>\n\n\n\n
Fraction = 5\/30. <\/p>\n\n\n\n
Simplify: GCF of 5 and 30 = 5. <\/p>\n\n\n\n
5\/30 = 1\/6.<\/p>\n\n\n\n
Answer: 1\/6<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 14<\/strong><\/p>\n\n\n\n
A school has 360 students. If 2\/5 of them play sport and 3\/4 of those who play sport also play a musical instrument, how many students play both sport and a musical instrument?<\/p>\n\n\n\n
Solution:<\/strong> Students who play sport = 2\/5 x 360 = 144. <\/p>\n\n\n\n
Students who play both = 3\/4 x 144 = 108.<\/p>\n\n\n\n
Answer: 108<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 15<\/strong><\/p>\n\n\n\n
Two fractions have a sum of 1. One fraction is 3\/7. What is the other fraction in simplest form?<\/p>\n\n\n\n
Solution:<\/strong> Other fraction = 1 – 3\/7 = 7\/7 – 3\/7 = 4\/7.<\/p>\n\n\n\n
Answer: 4\/7<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 16<\/strong><\/p>\n\n\n\n
Three fractions are in the ratio 1:2:3, and their sum is 1. What is the largest fraction?<\/p>\n\n\n\n
Solution:<\/strong> Let the fractions be k, 2k, and 3k. k + 2k + 3k = 1 <\/p>\n\n\n\n
6k = 1 k = 1\/6<\/p>\n\n\n\n
Largest fraction = 3k = 3\/6 = 1\/2.<\/p>\n\n\n\n
Answer: 1\/2<\/strong><\/p>\n\n\n\n
Key insight:<\/strong> When fractions are in a given ratio and their sum is known, introduce a variable k, write each fraction as a multiple of k, and set their sum equal to the known total.<\/p>\n\n\n\n
\n\n\n\nProblem 17<\/strong><\/p>\n\n\n\n
In a class of 40 students, the ratio of boys to girls is 3:5. How many girls are in the class?<\/p>\n\n\n\n
Solution:<\/strong> Total parts = 3 + 5 = 8. <\/p>\n\n\n\n
Girls = 5\/8 x 40 = 25.<\/p>\n\n\n\n
Answer: 25<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 18<\/strong><\/p>\n\n\n\n
A number is multiplied by 3\/4, and then the result is multiplied by 8\/9. The final result is 2. What was the original number?<\/p>\n\n\n\n
Solution:<\/strong> Let the number be \ud835\udc65. <\/p>\n\n\n\n
\ud835\udc65 x 3\/4 x 8\/9 = 2\ud835\udc65 x 24\/36 = 2\ud835\udc65 x 2\/3 = 2\ud835\udc65 = 3<\/p>\n\n\n\n
Verification:<\/strong> 3 x 3\/4 = 9\/4. 9\/4 x 8\/9 = 72\/36 = 2 \u2713<\/p>\n\n\n\n
Answer: 3<\/strong><\/p>\n\n\n\n
Key insight:<\/strong> Multiply the fractions together first before solving for \ud835\udc65 \u2014 3\/4 x 8\/9 = 24\/36 = 2\/3. This simplification step using equivalent fractions makes the arithmetic much cleaner.<\/p>\n\n\n\n
\n\n\n\nProblem 19<\/strong><\/p>\n\n\n\n
Which value of n makes the fractions 4\/7, n\/21, and 20\/35 all equivalent?<\/p>\n\n\n\n
Solution:<\/strong> Simplify 20\/35: GCF of 20 and 35 = 5. 20\/35 = 4\/7. So the target fraction is 4\/7.<\/p>\n\n\n\n
For n\/21: 21\/7 = 3. So n = 4 x 3 = 12.<\/p>\n\n\n\n
Verification:<\/strong> 4\/7 = 12\/21 = 20\/35. Cross multiply to check: 4 x 21 = 84, 7 x 12 = 84 \u2713<\/p>\n\n\n\n
Answer: n = 12<\/strong><\/p>\n\n\n\n
\n\n\n\nProblem 20 \u2014 hardest<\/strong><\/p>\n\n\n\n
The fraction (2a + 3)\/(3a + 5) equals 5\/7 for some value a. What is the value of 4a?<\/p>\n\n\n\n
Solution:<\/strong> Cross multiply: 7(2a + 3) = 5(3a + 5) <\/p>\n\n\n\n
14a + 21 = 15a + 25 <\/p>\n\n\n\n
14a – 15a = 25 – 21 -a = 4 a = -4<\/p>\n\n\n\n
4a = -16<\/p>\n\n\n\n
Verification:<\/strong> (2(-4) + 3)\/(3(-4) + 5) = (-8 + 3)\/(-12 + 5) = -5\/-7 = 5\/7 \u2713<\/p>\n\n\n\n
Answer: 4a = -16<\/strong><\/p>\n\n\n\n
Key insight:<\/strong> The question asks for 4a, not a \u2014 re-read the question before writing the final answer. This is a deliberate AMC technique to catch students who solve correctly but answer the wrong thing.<\/p>\n\n\n\n
\n\n\n\nEquivalent fractions reference sheet<\/strong><\/h2>\n\n\n\n
![\"Think](\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/04\/Screenshot-2026-04-14-at-12.30.53-PM.png\")
<\/a>