Grade 5 Adding and Subtracting Fractions

What this problem type covers

Eighteen problem types across five levels: direct addition and subtraction, three-term mixed-number calculations, checking a claim, combining quantities in context, converting fractional hours to hours and minutes, subtracting one overlap per join, finding a missing number, undoing a reversed operation, recovering an empty container mass, inclusion-exclusion with two groups, listing every natural number in a fraction inequality, maximizing a sum or difference made from number cards, decomposing a fraction into distinct unit fractions, recovering two fractions from their sum and difference, combining work rates, telescoping unit-fraction sums, finding a total from fractional shares and offsets, and finding water depth from a pole used at both ends.

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
   Calculate from left to right.
   2¹/₃ + 1³/₈ + 2¹/₄
   • ①5¹¹/₁₂
   • ②6¹/₂₄
   • ③3¹⁷/₂₄
   • ④5²³/₂₄
   • ⑤6
   Show solution

   ④5²³/₂₄

   1. Rewrite all three fractions with common denominators.
   2. The first two give 3¹⁷/₂₄.
   3. Applying the remaining operation gives 5²³/₂₄.
2. 2 · ★2
   5 strips, each 4²/₅ cm long, were joined with ³/₄ cm overlapping at each join.
   What is the total length?
   • ①22 cm
   • ②18¹/₄ cm
   • ③19 cm
   • ④20 cm
   • ⑤21¹/₄ cm
   Show solution

   ③19 cm

   1. Without overlap, the strips total 22 cm.
   2. There are 4 joins, so the overlap totals 3 cm.
   3. Subtracting gives 19 cm.
3. 3 · ★3
   A full container weighs 5 kg.
   After one quarter of the contents was removed, it weighed 4²/₃ kg.
   What does the empty container weigh?
   • ①3²/₃ kg
   • ②¹/₃ kg
   • ③1¹/₃ kg
   • ④4 kg
   • ⑤4²/₃ kg
   Show solution

   ①3²/₃ kg

   1. The difference, ¹/₃ kg, is one quarter of the contents.
   2. Multiplying by 4 gives 1¹/₃ kg of contents.
   3. Subtracting that from the full mass gives 3²/₃ kg.
4. 4 · ★4
   Use four of the cards 8, 6, 9, 7, 4 once each to make (proper fraction) + (proper fraction). Find the greatest possible result.
   • ①⁸/₉
   • ②1⁷/₉
   • ③1¹⁶/₂₁
   • ④1⁴⁷/₆₃
   • ⑤⁶/₇
   Show solution

   ④1⁴⁷/₆₃

   1. Check every choice of four cards and every numerator-denominator arrangement.
   2. The greatest arrangement is ⁸/₉ + ⁶/₇.
   3. It equals 1⁴⁷/₆₃.
5. 5 · ★5
   A 10⁶/₇ m pole was lowered to the bottom from one end, removed, and lowered again from the other end.
   After both dips, 2⁴/₇ m of the pole was still dry.
   How deep is the water?
   • ①4²/₇ m
   • ②6⁵/₇ m
   • ③2⁴/₇ m
   • ④8²/₇ m
   • ⑤4¹/₇ m
   Show solution

   ⑤4¹/₇ m

   1. Removing the dry part leaves 8²/₇ m wet.
   2. Using both ends makes the wet length twice the depth.
   3. Halving it gives a depth of 4¹/₇ m.

Where people usually go wrong

• Adding denominators as well as numerators. Unlike denominators must be rewritten before the numerators can be combined.
• Subtracting mixed numbers without regrouping when the first fractional part is smaller.
• Treating a fractional hour as the same number of minutes. A quarter hour is 15 minutes, not 25 minutes.
• Counting one overlap for every strip. A row of four strips has three joins.
• Undoing a mistaken operation once. Replacing subtraction with addition changes the result by two copies of the operand.
• Adding the sizes of two groups without removing the intersection that was counted twice.
• Stopping after one promising number-card arrangement. A greatest-value claim needs every valid arrangement checked.
• Adding every term in a telescoping sum separately and missing the cancellation of the middle unit fractions.