Depth · one concept from first principles · step 2 of 4
Same numerator: smaller denominator is larger
Find the values of and that satisfy both of the following equations.
Fractions
Your answer
How to solve
Strategy Guess and Check — Each equation says one fraction equals a simple unit fraction, so an equivalent fraction with numerator A must match. We test small whole-number values of A, build the matching denominators, and check which single A makes both denominators differ by exactly the right amount. This is sturdier than jumping to algebra and lets a young learner reason from equivalent fractions they already know.
1STEP 1
Turn each fraction equation into a denominator rule
A over (B+4) equalling one fourth means the bottom is 4 copies of the top: B + 4 = 4 x A.
B+4 = 4 × A B+9 = 5 × A
An equivalent fraction keeps the same value: to get 1/4 from A/(B+4), the bottom is just A copied 4 times. Fourth graders already build equivalent fractions this way.
3.NF.A.3Identify SubproblemsA over (B+4) being one fourth means B+4 is 4 times A.
Why?
Two fractions are equal when one is the other with the top and the bottom scaled by the same number, so A over (B+4) is one fourth with both parts multiplied by A.
Why?
A written over A is one, and multiplying one fourth by one leaves it unchanged.
🧱Multiplicative identity (one)One group of a number is just that number — multiplying by one changes nothing.
Why?
Scaling the 1 and the 4 by A puts A on top and four groups of A on the bottom.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Guess values of A and build both denominators
Trying whole numbers, the two rules hand back the same B only when A is 5.
A=5: B = 4×5-4 = 16 and B = 5×5-9 = 16
Testing small numbers and watching the two B-values close in on each other is a concrete way to find the unknown without abstract symbol pushing.
3.OA.A.4Guess And Check3STEP 3
See the jump pattern and confirm A=5
The denominators differ by 5, and moving from a fourth to a fifth adds one copy of A, so A = 5.
(B+9)-(B+4)=5 × A-4 × A = A = 5
Seeing that adding 5 to the bottom is exactly 'one more copy of A' reveals the answer as a comparison, confirming the guess instead of relying on luck.
4.OA.A.1Look For A Pattern4STEP 4
Find B and double-check both equations
With A = 5 the first rule gives B = 16, and 5/20 and 5/25 do reduce to a fourth and a fifth.
5/20=1/4 5/25=1/5
Plugging the found numbers back in and simplifying confirms the fractions really do equal 1/4 and 1/5.
3.NF.A.3Guess And CheckAnswer
A = 5, B = 16
Both checks work: 5/20 reduces to 1/4 and 5/25 reduces to 1/5, and the same A=5, B=16 satisfy both sentences. The denominators 20 and 25 differ by 5, exactly the value of A, which is the pattern we used.
Takeaway
When a fraction equals a unit fraction like 1/4, the bottom is just the top copied 4 times, so you can test small numbers until both sentences agree.
- Turn each fraction equation into a denominator rule
- Guess values of A and build both denominators
- See the jump pattern and confirm A=5
- Find B and double-check both equations
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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The same shape shows up later on the AMC, as Small-Domain Constrained Search. Why the shapes repeat →