Grade 5 Simplifying Fractions and Common Denominators

What this problem type covers

Twenty-one problem types across five levels: missing terms in equivalent fractions, simplifying to lowest terms, finding or checking common denominators, comparing fractions, listing fractions between two bounds, finding the closest fraction with a fixed denominator, counting equivalent fractions in a denominator range, rebuilding a fraction from a sum, difference, or product, finding distant terms in a fraction sequence before rewriting them with a common denominator, reversing numerator and denominator changes, adding the same number to both terms, solving two same-numerator fraction equations, bounding an unknown denominator, counting reduced or reducible fractions, filtering an interval for fractions in lowest terms, choosing an extreme fraction from number cards, adding fractions that reduce to unit fractions, locating a reducible fraction in a sequence, rebuilding a ratio from the least common multiple of its terms, intersecting numerator, denominator, and parity conditions, and completing a denominator to a square or cube.

Example problems

One example from each difficulty band, with worked solutions.

These problems are generated automatically. The answers are derived as each problem is built and are checked by tests, but mistakes are still possible — please tell us if you find one.

1. 1 · ★1
   Write ³⁶/₈₄ in lowest terms.
   • ①³⁶/₇
   • ②³/₈₄
   • ③⁷/₃
   • ④⁴/₇
   • ⑤³/₇
   Show solution

   ⑤³/₇

   1. The greatest common divisor of 36 and 84 is 12.
   2. Divide both terms by 12.
   3. The fraction in lowest terms is ³/₇.
2. 2 · ★2
   Find the largest fraction with denominator 35 strictly between ¹²/₃₅ and ¹⁶/₃₅.
   • ①¹⁵/₃₅
   • ②¹⁴/₃₅
   • ③¹³/₃₅
   • ④¹²/₃₅
   • ⑤¹⁶/₃₅
   Show solution

   ①¹⁵/₃₅

   1. Rewrite both bounds with denominator 35.
   2. The valid numerators run from 13 through 15.
   3. Choosing the largest numerator gives ¹⁵/₃₅.
3. 3 · ★3
   The same whole number is added to both terms of ⁴⁰/₁₁₀, and the result reduces to ⁵/₁₂. What was added?
   • ①8
   • ②70
   • ③9
   • ④20
   • ⑤10
   Show solution

   ⑤10

   1. After adding, the unreduced fraction is ⁵⁰/₁₂₀.
   2. The numerator increased by 10.
   3. The denominator increased by the same amount.
4. 4 · ★4
   Add every proper fraction with denominator 30 that reduces to a unit fraction.
   • ①⁸/₅
   • ②⁷/₅
   • ③⁴²/₃₀
   • ④⁷/₃₀
   • ⑤⁴¹/₃₀
   Show solution

   ②⁷/₅

   1. The numerator must be a proper divisor of 30.
   2. Those numerators are 1, 2, 3, 5, 6, 10, 15.
   3. Their sum gives ⁷/₅.
5. 5 · ★5
   Find the fraction whose numerator and denominator have least common multiple 288 and whose lowest terms are ³/₈.
   • ①⁹⁶/₃₆
   • ②³⁶/₉₆
   • ③³/₈
   • ④³⁷/₉₆
   • ⑤³⁹/₁₀₄
   Show solution

   ②³⁶/₉₆

   1. Write the terms as 3 × k and 8 × k.
   2. Their least common multiple gives k = 12.
   3. The fraction is ³⁶/₉₆.

Where people usually go wrong

• Multiplying the denominator but leaving the numerator unchanged. An equivalent fraction must scale both terms by the same factor.
• Calling a fraction reduced after dividing by one common factor. Lowest terms means the two remaining terms have no common factor greater than 1.
• Using the product of two denominators as the only possible common denominator. Every common multiple works, though the least one is usually easiest.
• Comparing denominators directly when the numerators are different. Rewrite the fractions with a common numerator or denominator first.
• Including an endpoint when the wording says strictly between. The two boundary fractions are not part of the answer.
• Rounding toward only one neighboring numerator in a closest-fraction problem. Both neighbors must be checked because either one can be closer.
• Counting every numerator in a range as a fraction in lowest terms. Each numerator still has to be checked against the fixed denominator.
• Reversing only the last numerator change in a backward problem and forgetting the denominator change.