Problem
Consecutive integers around a middle value
Three whole numbers in a row (each one bigger than the last) add up to 27. Find the product of those three numbers.
Operations
Your answer
Watch out Finding the middle number, 9, is only step two — the question asks for the product of all three numbers, not the middle number itself.
How to solve
Strategy Look for a Pattern — Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.
1STEP 1
The sum is 3 times the middle number
Write the numbers around the middle: middle-1, middle, middle+1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.
(m-1) + m + (m+1) = 3 × m = 27
Evenly spaced numbers balance around their center, so their sum is the center times how many there are.
3.OA.D.9Look For A PatternThe sum of three consecutive whole numbers is exactly 3 times the middle number.
Why?
The smallest number is 1 below the middle and the largest is 1 above, so sliding that single unit from the largest down to the smallest makes all three equal to the middle while the total stays the same.
Why?
Moving 1 unit from one number to another only rearranges the same total among the three numbers, so the sum does not change.
Why?
However you split a fixed total into parts, those parts always add back to that same total.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Once all three are the middle number, the sum is middle plus middle plus middle, which is three equal groups of the middle.
Why?
Adding the same amount three times is the same as taking 3 groups of it, which is 3 times that amount.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Find the middle number
Since 3 times the middle is 27, divide to get the middle number: 27 ÷ 3 = 9.
m = 27 ÷ 3 = 9
Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.
3.OA.C.7Guess And Check3STEP 3
List the numbers and multiply
The numbers are 8, 9, 10 (check: 8 + 9 + 10 = 27). Multiply them together.
8 × 9 × 10 = 72 × 10 = 720
Multiply two at a time; times 10 just appends a zero.
3.OA.C.7Look For A PatternAnswer
720
8 × 9 × 10 = 72 × 10 = 720
8 + 9 + 10 = 27 matches the given sum. The product 720 is close to 9 cubed (729), which makes sense for three numbers near 9.
Takeaway
Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!
- The sum is 3 times the middle number
- Find the middle number
- List the numbers and multiply
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested