Problem
A denominator that cancels to 1 leaves a whole number
One improper fraction divides two given fractions and gives a whole number both times. We need the smallest such fraction.
Number system
Your answer
How to solve
Strategy Convert to Algebra — Each condition, once the division is flipped into a multiplication, says something separate about the numerator and about the denominator. Collect those four demands and the smallest fraction is the one with the smallest allowed numerator over the largest allowed denominator.
1STEP 1
Flip both divisions
Call the fraction ■/▲. Dividing by 3/8 multiplies by 8/3, and dividing by 7/6 multiplies by 6/7.
1 1/6 = 7/6
Two products have to come out whole.
6.NS.A.1Convert To Algebra2STEP 2
Read what each product demands
For 8■/3▲ to be whole, the 3 has to be cancelled by ■ and ▲ has to divide 8. The other product asks the same of 7 and 6.
Four demands: 3 and 7 divide the top; the bottom divides 8 and 6.
6.NS.B.4Eliminate PossibilitiesFor the product to be a whole number the numerator must carry the factors the denominator needs.
Why?
A fraction is a whole number exactly when the bottom divides the top with nothing left over, because that is what makes the division come out even.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Multiplying by puts a underneath, so a factor of has to be waiting in the numerator to cancel it away.
🧱Multiplicative identity (one)One group of a number is just that number — multiplying by one changes nothing.
3STEP 3
Find the largest possible denominator
The denominator has to divide both 8 and 6, so it is a common factor of the two, and the largest is their greatest common factor.
8 = 2 × 4, 6 = 2 × 3
The denominator can be 2 at most.
6.NS.B.4Make A Systematic List4STEP 4
Find the smallest possible numerator
The numerator has to be a multiple of both 3 and 7, so the smallest is their least common multiple.
3 × 7 = 21
The fraction is 21/2.
6.NS.B.4Make A Systematic ListAnswer
Test both conditions: 21/2 ÷ 3/8 = 28 and 21/2 ÷ 7/6 = 9, both whole numbers.
Takeaway
A product comes out whole when the numerator carries the factors the denominator needs -- read that demand for the top and bottom separately.
- Flip both divisions
- Read what each product demands
- Find the largest possible denominator
- Find the smallest possible numerator