Grade 4 Adding and Subtracting Fractions

What this problem type covers

Eighteen problem types across five levels: undoing a calculation that added where it should have subtracted, recovering two fractions from their sum and difference, a rectangle whose sides differ by a fraction, weighing a box with books taken out, tape strips joined with an overlap at each join, a made-up rule applied several times over, a pole whose wet part covers the depth twice, a candle burning by a fixed amount every so many minutes, a clock that gains a few sixtieths of a minute a day, the largest and smallest mixed numbers from a hand of number cards, the whole numbers strictly between two results, three weights known only two at a time, a run whose whole part falls and whose numerator climbs, adding every proper fraction over one denominator, a job finished with help partway through, three fractions tied by a difference and a doubling, and a distance along a line where two stretches overlap.

Example problems

One example from each difficulty band, with worked solutions.

Estos problemas se generan automáticamente. Las respuestas se derivan al crear cada problema y se comprueban con pruebas, pero puede haber errores: avísanos si encuentras alguno.

1. 1 · ★1
   Two proper fractions have denominator 8. They add to 1 and differ by ²/₈. What is the larger one?
   • ①⁴/₈
   • ②³/₈
   • ③⁵/₈
   • ④⁶/₈
   • ⑤1²/₈
   Show solution

   ③⁵/₈

   1. Adding the sum and the difference gives twice the larger number.
   2. 1 + ²/₈ is twice the larger.
   3. Halving it gives ⁵/₈.
   4. Taking that from the sum leaves ³/₈.
2. 2 · ★2
   A box holding 5 identical books weighs 8 kg. With only 2 books left in it, the box weighs 4²/₈ kg. How much does one book weigh?
   • ①1²/₈
   • ②1⁷/₈
   • ③3⁶/₈
   • ④⁶/₈
   • ⑤1³/₈
   Show solution

   ①1²/₈

   1. The difference between the two weighings is the books taken out.
   2. That difference is 3⁶/₈ kg, for 3 books.
   3. Divide 3⁶/₈ by 3.
   4. One book weighs 1²/₈ kg.
3. 3 · ★3
   A pole 16⁶/₁₅ m long is put straight down to the bottom of a pond, pulled out, turned end for end and put down again. The part still dry is 6⁴/₁₅ m. How deep is the pond?
   • ①10²/₁₅
   • ②11⁵/₁₅
   • ③5²/₁₅
   • ④5¹/₁₅
   • ⑤6⁴/₁₅
   Show solution

   ④5¹/₁₅

   1. Turning the pole means the wet part covers the depth twice.
   2. The pole less the dry part is 10²/₁₅ m.
   3. That is twice the depth.
   4. The pond is 5¹/₁₅ m deep.
4. 4 · ★4
   Using each of the cards 3, 6, 5, 7 once, two mixed numbers with denominator 15 are added. What is the largest possible total?
   • ①12⁹/₁₅
   • ②13⁷/₁₅
   • ③13⁸/₁₅
   • ④8¹³/₁₅
   • ⑤13⁹/₁₅
   Show solution

   ③13⁸/₁₅

   1. To make the sum largest, put the big cards in the whole-number parts.
   2. The two largest give whole parts adding to 13.
   3. The other two become numerators, adding to 8.
   4. The largest total is 13⁸/₁₅.
5. 5 · ★5
   Numbers are listed by a rule. What do they add to?
   15¹/₁₈, 13³/₁₈, 11⁵/₁₈, …, 1¹⁵/₁₈
   • ①69⁷/₁₈
   • ②64
   • ③67¹¹/₁₈
   • ④65¹³/₁₈
   • ⑤67¹⁰/₁₈
   Show solution

   ⑤67¹⁰/₁₈

   1. The whole parts fall by 2 and the numerators rise by 2, so there are 8 terms.
   2. The whole parts add to 64.
   3. The numerators add to ⁶⁴/₁₈.
   4. As a mixed number that is 3¹⁰/₁₈.
   5. Added to the whole parts' total of 64, that gives 67¹⁰/₁₈.

Where people usually go wrong

• Undoing a wrongly added number once instead of twice.
• Counting an overlap for every strip rather than for every join.
• Rounding the wire down when the leftover still needs a whole metre.
• Reading the sixtieths of a minute as seconds without converting.
• Subtracting the depth once when the pole covers it going down and coming back up.
• Reducing a fraction that the answer is meant to keep as it is.
• Including the two end values when the question says strictly between.
• Adding all three given stretches, which counts the overlap twice.