Depth · one concept from first principles · step 2 of 4
Smallest fraction: LCM of denominators over GCF
A number times 1 7/8 is a whole number, and the same number divided by 5/12 is a whole number. Find the smallest such number.
Number system
Your answer
How to solve
Strategy Identify Subproblems — Turn both conditions into multiplications by a fraction. Then write the unknown as a fraction in lowest terms and ask what its numerator and denominator must do for each product to come out whole.
1STEP 1
Make both conditions multiplications
Change the mixed number to an improper fraction, and replace dividing by 5/12 with multiplying by its reciprocal.
1 7/8 = 15/8, ÷ 5/12 = × 12/5
Now the number times 15/8 and times 12/5 must both be whole.
7.NS.A.2Identify Subproblems2STEP 2
Find what the numerator and denominator need
Write the number as p/q in lowest terms. For both products to be whole, p must be a multiple of 8 and 5, and q must divide 15 and 12.
p: multiple of 8 and 5, q: divisor of 15 and 12
The numerator is a common multiple; the denominator is a common divisor.
6.NS.B.4Convert To AlgebraWriting the number as in lowest terms, must be a multiple of both and , and must divide both and .
Why?
A fraction comes out whole only when every prime on the bottom is cancelled by the same prime on the top, and each number has only one prime breakdown to check.
🧱Every number has one prime factorizationKeep splitting a number into factors and you always land on the same bag of primes, no matter which split you tried first.
Why?
In the three s of find no in , so all of them must come from ; in the finds no in , so it too must come from .
🧱Every number has one prime factorizationKeep splitting a number into factors and you always land on the same bag of primes, no matter which split you tried first.
Why?
and share no prime because the fraction is in lowest terms, so every prime of must be cancelled by and also by , which means divides both.
🧱Every number has one prime factorizationKeep splitting a number into factors and you always land on the same bag of primes, no matter which split you tried first.
3STEP 3
Make the fraction as small as possible
Smallest fraction: smallest numerator, largest denominator, so the LCM of 8 and 5 over the GCF of 15 and 12.
40/3
The LCM of 8 and 5 is 40; the GCF of 15 and 12 is 3.
6.NS.B.4Identify SubproblemsAnswer
40/3
Check both products: 40/3 × 15/8 = 600/24 = 25 and 40/3 × 12/5 = 480/15 = 32, both natural numbers.
Takeaway
To make two fraction products whole, the numerator collects the LCM and the denominator keeps the GCF.
- Make both conditions multiplications
- Find what the numerator and denominator need
- Make the fraction as small as possible
Where next?
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