Depth · one concept from first principles · step 2 of 4
Find the extreme decimal satisfying an inequality
Find the greatest number with two decimal places that can fill the so that + 4.17 stays less than 10.45 - 2.71.
Base-ten numbersFractions
Your answer
Watch out Treating the inequality as 'less than or equal to' and stopping at 3.57, when the must be strictly less, so the answer is one hundredth smaller.
How to solve
Strategy Work Backwards — First simplify the right side to a single number, then work backwards to isolate the . The must be less than some value, so the greatest two-decimal number is the largest hundredths value strictly below it.
1STEP 1
Simplify the right side
Compute the right-hand subtraction: 10.45 - 2.71 = 7.74. So the inequality becomes + 4.17 less than 7.74.
10.45 - 2.71 = 7.74
Turn the right side into one clean number before solving.
5.NBT.B.7Work Backwards2STEP 2
Isolate the
Subtract 4.17 from both sides to undo the addition: less than 7.74 - 4.17 = 3.57.
Undo the +4.17 by subtracting it, leaving the alone.
5.NBT.B.7Work Backwards3STEP 3
Pick the greatest two-decimal number below 3.57
The must be strictly less than 3.57. The largest hundredths number below 3.57 is one hundredth less: 3.56.
□ = 3.57 - 0.01 = 3.56
Since it can't equal 3.57, step down one hundredth to the biggest allowed value.
4.NF.C.7Guess And CheckOnce the must stay below 3.57, the greatest two-decimal-place number that fits is the two-decimal value exactly one hundredth beneath that limit.
Why?
Stepping one hundredth down from 3.57 lands on a two-decimal number that really is smaller than the limit, so it is an allowed choice.
Why?
That number shares the same ones and the same tenths as 3.57 but holds one fewer hundredth, so lining up the places shows it sits below 3.57.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
No two-decimal number that is still below 3.57 can be any larger, because the very next two-decimal value above it is 3.57 itself, which the may not reach.
Why?
Two-decimal numbers move one hundredth at a time with nothing in between, so one hundredth above the chosen value is already 3.57 and no two-decimal number is skipped over.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Answer
3.56
□ = 3.57 - 0.01 = 3.56
Test 3.56: 3.56 + 4.17 = 7.73, which is less than 7.74 - true. Test 3.57: 3.57 + 4.17 = 7.74, not less than 7.74 - false. So 3.56 is indeed the greatest value that works.
Takeaway
Clean up each side, undo the addition to free the , then step down one hundredth for the biggest 'less-than' answer!
- Simplify the right side
- Isolate the
- Pick the greatest two-decimal number below 3.57
Where next?
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A parent dashboard for the family lives at sensimlab.com.
The same shape shows up later on the AMC, as Multiplicative Ratio with Integrality Constraints, Small-Domain Constrained Search. Why the shapes repeat →