Reasoning · Grade 3-2 Size of Fractions

Problem

Build fractions from digit cards

Use three of the cards 2, 3, 6, 8 once each to build a fraction. Three cards means one part is two digits and the other is one. An improper fraction needs the numerator to be at least the denominator. Find every improper fraction under 10.
Your answer
How to solve
Strategy Make a Systematic List — Because only a few cards are involved, the surest way to catch ALL the fractions is to list every legal arrangement in an organized way, then check each one against the two rules (improper, and less than 10).
1STEP 1

Decide the shape of the fraction

Improper means the two-digit part is the numerator.

(two-digit)/(one-digit)
2STEP 2

List by choosing the single denominator card

With denominator 8 all six stay under 10.

82 ÷ 6 = 13.66 ≥ 10
3STEP 3

Finish the list with the small denominators

Denominator 6 keeps four and denominator 3 only two.

26 ÷ 3 = 8.66 ＜ 10 62 ÷ 3 ≈ 20.7 ≥ 10
4STEP 4

Write the complete list

Denominator 2 fails entirely, leaving 12 in all.

Answer
12 fractions
23/8, 26/8, 23/6, 32/8, 36/8, 28/6, 32/6, 38/6, 62/8, 63/8, 26/3, 28/3
Each listed fraction has a two-digit top and one-digit bottom, so it is genuinely improper; computing each value gives results between about 2.9 and 9.3, all under 10, and the ones we dropped (like 82/6 = 13.67) really are 10 or more. The count of 12 matches a careful re-scan of the organized list.
Takeaway

List every choice in a tidy order and you will never miss a fraction or count one twice!

  • Decide the shape of the fraction
  • List by choosing the single denominator card
  • Finish the list with the small denominators
  • Write the complete list