Operations & Word Problems

Problem

Solve shape-coded products in order

Each shape stands for the same number. ●, ▲, and ■ are three different one-digit numbers. Given ● × ● = 36, ▲ × ● = 48, and ■ × 2 = 1▲ (the two-digit number with tens digit 1 and ones digit ▲), find ● + ▲ + ■.
Operations
Your answer
How to solve
Strategy Eliminate Possibilities — Solve in the order that pins down one shape at a time: ● first from the squared equation, then ▲, then ■.
1STEP 1

Find ●

Only 6 times itself makes 36 among one-digit numbers.

6 × 6 = 36
2STEP 2

Find ▲

▲ × 6 = 48, so ▲ is 8.

▲ × 6 = 48 → ▲ = 8
3STEP 3

Find ■ and the sum

1▲ is 18, so ■ × 2 = 18 gives ■ = 9; adding the three gives 23.

■ × 2 = 18 → ■ = 9, 6 + 8 + 9 = 23
Answer
23
6 + 8 + 9 = 23
Check — 6×6=36, 8×6=48, and 9×2=18 (=1▲ with ▲=8) all hold, so 6+8+9=23.
Takeaway

Solve the equation that pins one shape down first, then the rest fall into place.

  • ● × ● = 36 → ● = 6
  • ▲ × 6 = 48 → ▲ = 8
  • 1▲=18 → ■ = 9, sum 23
Where next?
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