Problem
Scaling numerator and denominator keeps value
I need a fraction that reduces to 4/7. Among all such fractions, I want the one whose denominator is exactly 15 more than its numerator.
Fractions
Your answer
How to solve
Strategy Look for a Pattern — Every fraction equal to 4/7 is built by multiplying both 4 and 7 by the same number. That makes a clear pattern: 4/7, 8/14, 12/21, ... Looking at how the gap between top and bottom grows as I scale lets me find the right multiplier directly, and I can guess-and-check the multiplier to confirm the gap is 15.
1STEP 1
List equivalent fractions
Multiplying top and bottom by the same number keeps 4/7's value, giving 8/14, 12/21 and so on.
4/7, 8/14, 12/21, 16/28, 20/35
A fourth grader can make equal fractions just by multiplying top and bottom by the same number, like making a bigger but identical recipe.
4.NF.A.1Look For A Pattern2STEP 2
Track the gap between bottom and top
Subtracting top from bottom gives 3, 6, 9, so the gap is always 3 times the multiplier.
7k-4k=3k
Since both numbers are scaled by the same multiplier, their difference scales too, so the gap follows an easy times-3 pattern.
3.NF.A.3Look For A PatternIn every fraction equal to 4/7, the denominator minus the numerator is 3 times the multiplier used.
Why?
Multiplying the numerator and the denominator by the same number leaves the fraction's value untouched, so all of these really are 4/7.
Why?
A number written over itself is one, and multiplying by one changes nothing.
🧱Multiplicative identity (one)One group of a number is just that number — multiplying by one changes nothing.
Why?
Since both parts are stretched by the same multiplier, the distance between them is stretched by that multiplier too, starting from the original gap of 3.
Why?
The denominator is the numerator together with the gap, so multiplying the whole thing multiplies the numerator part and the gap part alike.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
3STEP 3
Find the multiplier that gives a gap of 15
A gap of 15 therefore needs a multiplier of 15 divided by 3 = 5.
3k=15 → k=5
Counting up by 3s (3, 6, 9, 12, 15) lands on 15 at the fifth step, so multiplying by 5 is the answer.
4.NF.A.1Guess And Check4STEP 4
Build the fraction
Multiply 4 and 7 by 5 to get the numerator and denominator.
(4 × 5)/(7 × 5)=20/35
Once the multiplier is known, the fraction comes straight from scaling 4 and 7 by it.
4.NF.A.1Guess And CheckAnswer
Check the two conditions: 35 minus 20 is 15, and 20/35 divides by 5 on top and bottom to give 4/7. Both conditions hold, so 20/35 is correct.
Takeaway
Equal fractions come from multiplying top and bottom by the same number, and the gap between them grows by that same factor.
- List equivalent fractions
- Track the gap between bottom and top
- Find the multiplier that gives a gap of 15
- Build the fraction
Where next?
Another one like thissuggested
Step back and solidifysuggested