Depth · one concept from first principles · step 2 of 4
An improper fraction divided by and the same fraction divided by both give whole numbers. Find the smallest such improper fraction.
Call the fraction ■/(▲). Dividing by 3/8 multiplies by 8/3, and dividing by 7/6 multiplies by 6/7.
Two products have to come out whole.
6.NS.A.1Convert To AlgebraFor 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.
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.
Multiplying by puts a underneath, so a factor of has to be waiting in the numerator to cancel it away.
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.
The denominator can be 2 at most.
6.NS.B.4Make A Systematic ListThe numerator has to be a multiple of both 3 and 7, so the smallest is their least common multiple.
The fraction is 21/2.
6.NS.B.4Make A Systematic ListA product comes out whole when the numerator carries the factors the denominator needs -- read that demand for the top and bottom separately.
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