Depth · one concept from first principles · step 2 of 4
Find the rule and write it as an expression
Pairs of numbers each produce one result by the same hidden rule. We need the result for the last pair.
ExpressionsOperationsNumber system
Your answer
How to solve
Strategy Look for a Pattern — Guess a relationship from the first pair, then test it on the others. A rule that survives all three examples is worth trusting; one that fits only the first is a coincidence.
1STEP 1
Look for what the two numbers do together
The results are much smaller than the pairs, so try adding the two numbers and shrinking: the first pair sums to 12 and gives 2.4.
4 + 8 = 12
12 becomes 2.4, which is a fifth of it.
5.OA.B.3Guess And Check2STEP 2
Test the guess on the other pairs
Add each pair and divide by 5: the second gives 19 ÷ 5 and the third 43 ÷ 5, and both match the printed results.
19 ÷ 5 = 3.8, 43 ÷ 5 = 8.6
The rule holds for all three.
6.NS.B.3Look For A PatternThe rule guessed from the first pair really is the rule.
Why?
A rule has to give the printed result for every pair, not just for one, so a guess is only worth trusting once the other pairs agree with it.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Adding the two numbers and taking a fifth gives , and in turn, and each of those is the result the page prints beside that pair.
🧱Transitivity of equalityIf the first equals the second and the second equals the third, the first equals the third.
3STEP 3
Write the rule down
Whatever the pair is, add the two numbers and divide the sum by 5.
(first + second) ÷ 5
One rule covers every pair.
6.EE.A.2Look For A Pattern4STEP 4
Use it on the last pair
Add 9 and 25, then divide by 5.
9 + 25 = 34, 34 ÷ 5 = 6.8
The empty space holds 6.8.
6.NS.B.3Solve An Easier Related ProblemAnswer
6.8
Every result is a fifth of its pair's sum: 12, 19, 43 and 34 give 2.4, 3.8, 8.6 and 6.8.
Takeaway
Test a guessed rule on every example you have -- one that fits only the first is a coincidence.
- Look for what the two numbers do together
- Test the guess on the other pairs
- Write the rule down
- Use it on the last pair
Where next?
Level upsuggested
Another one like thissuggested
Same look, different logicsuggested
Step back and solidifysuggested
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Geometric Series Share, Lattice / Grid Pattern from Small Cases. Why the shapes repeat →