Problem
A new symbol can define a custom operation
A new symbol is defined so that a ◆ b means a times b, then plus 30. Using this rule, first compute 32 ◆ 15, then apply ◆ 23 to that result to get (32 ◆ 15) ◆ 23.
Operations
Your answer
Watch out Don't stop at 510 × 23 = 11730 — the ◆ rule means add 30 again for the outer step too.
How to solve
Strategy Identify Subproblems — The expression has a parentheses-first structure, so we split it into two subproblems: first evaluate the inner 32 ◆ 15, then feed that result into a second application of the same rule with 23. Each application follows the identical pattern multiply-then-add-30.
1STEP 1
Evaluate the inner operation
Apply the rule to 32 and 15: 32 × 15 = 480, then 480 + 30 = 510.
32 ◆ 15 = 32 × 15 + 30 = 480 + 30 = 510
The parentheses tell us to finish the inside first, exactly like order of operations says.
5.OA.A.1Identify SubproblemsThe inner operation 32 ◆ 15 works out to 510.
Why?
The symbol ◆ is defined as multiply the two numbers then add 30, so 32 ◆ 15 stands for 32 × 15 + 30, and finding its value is just carrying out that multiply-then-add recipe.
Why?
Multiplying 32 by 15 gives 480, and multiplying simply gathers 32 equal groups of 15 into one amount.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Adding 30 to that 480 merges two separate amounts into a single total of 510, with nothing missing and nothing counted twice.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Evaluate the outer operation
Apply the same rule to 510 and 23: 510 × 23 = 11730, then 11730 + 30 = 11760.
510 ◆ 23 = 510 × 23 + 30 = 11730 + 30 = 11760
The custom symbol is just a recipe; reusing the same multiply-then-add-30 pattern finishes the job.
4.OA.A.3Look For A PatternAnswer
11760
510 ◆ 23 = 510 × 23 + 30 = 11730 + 30 = 11760
The answer is dominated by 510 × 23 = 11730; adding 30 nudges it to 11760, which is sensibly just above 11730. Both steps follow the defined rule exactly, and the inside-first ordering was respected.
Takeaway
A made-up symbol is just a recipe -- do the inside first, then follow the same multiply-and-add-30 steps again!
- Evaluate the inner operation
- Evaluate the outer operation
Where next?
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