Operations & Word Problems

Problem

Both sides of an equation can be multiplied or divided by the same number

Two expressions, each built from one unknown, give the same result. We need the value of the first unknown divided by the second.
Expressions
Your answer
How to solve
Strategy Work Backwards — Simplify each side into a single multiple of its unknown. Then the equality is one clean sentence relating the two, and the ratio falls out without ever needing the individual values.
1STEP 1

Simplify the first expression

Dividing by 2 and then multiplying by 6 is the same as multiplying by 6 ÷ 2.

○ ÷ 2 × 6 = ○ × 3
2STEP 2

Simplify the second expression

Dividing by 3 twice is dividing by 9 once.

△ ÷ 3 ÷ 3 = △ ÷ 9
3STEP 3

Set the two sides equal

The problem says the results match, so write that as one equation and clear the division by multiplying both sides by 9.

○ × 3 = △ ÷ 9 → △ = ○ × 27
4STEP 4

Read off the ratio

If △ is 27 copies of ○, then ○ is one twenty-seventh of △, whatever the actual numbers are.

○ ÷ △ = 1/27
Answer
127\frac{1}{27}
Try a case: with ○ = 1 the first side is 3, and △ must be 27 because 27 ÷ 9 = 3. Then 1 ÷ 27 = 1/27.
Takeaway

Tidy each side on its own first -- an equation is much easier to read once both sides are as simple as they can be.

  • Simplify the first expression
  • Simplify the second expression
  • Set the two sides equal
  • Read off the ratio

▶ Practice — 12 problems