Reasoning · Grade 3-2 Size of Fractions

Problem

Equivalent fractions from a diagram

Adding the same number to the top and bottom of 5/9 makes it equal 9/10. The added number is the same above and below. Adding equally keeps the denominator-minus-numerator gap unchanged. Find the number added.
Your answer
How to solve
Strategy Guess and Check — Since 5/9 must become a fraction equal to 9/10, we can list fractions equal to 9/10 (the equivalent-fraction pattern) and check which one could come from 5/9 by adding the same number to top and bottom. A key clue speeds this up: adding the same number to both keeps the gap between bottom and top fixed.
1STEP 1

Notice the gap stays the same

5/9 has a gap of 4, and that gap never changes.

9 - 5 = 4
2STEP 2

List fractions equal to 9/10

Among fractions equal to 9/10, the one with gap 4 is 36/40.

9/10 = 18/20 = 27/30 = 36/40 = 45/50
3STEP 3

Pick the one with gap 4 and read off the added number

Going from 5 to 36, the number added is 31.

(5+31)/(9+31) = 36/40 = 9/10
Answer
31
36/40 = 9/10
Check directly: (5+31)/(9+31) = 36/40, and 36/40 divided top and bottom by 4 is 9/10, so it really equals 9/10. Both numerator and denominator rose by the same 31, as required.
Takeaway

Adding the same number to the top and bottom keeps their gap the same, so just find the matching equivalent fraction!

  • Notice the gap stays the same
  • List fractions equal to 9/10
  • Pick the one with gap 4 and read off the added number