Real problems mix fractions, decimals and percents, and take several steps. You have every tool already: percents in stores, equations from stories and all four operations. The new part is planning and checking.
Key idea
Read, estimate, then solve one step at a time. Switch between fractions, decimals and percents when it makes a step easier ($25\% = \frac{1}{4} = 0.25$). Compare your answer with the estimate at the end.
Why it works
An estimate is a quick, rough answer you make before the exact one. Most slips in multistep problems, like a misplaced decimal point or a percent used as a whole number, make the answer far from the estimate, so the estimate catches them.
| Fraction | Decimal | Percent |
|---|---|---|
| $\frac{1}{2}$ | $0.5$ | $50\%$ |
| $\frac{1}{4}$ | $0.25$ | $25\%$ |
| $\frac{3}{4}$ | $0.75$ | $75\%$ |
| $\frac{1}{5}$ | $0.2$ | $20\%$ |
| $\frac{1}{10}$ | $0.1$ | $10\%$ |
Worked examples
Worked example: Discount, then tax
A \$45 jacket is 20% off, and then 6% sales tax is added. What is the total?
- Estimate: 20% off \$45 is about \$36, plus a little tax, so about \$38.
- Sale price: $0.80 \times 45 = 36$.
- With tax: $1.06 \times 36 = 38.16$. Close to the estimate ✓.
Answer: \$38.16
Worked example: A raise
Omar earns \$15 an hour and gets a 10% raise. How much does he earn for 30 hours at the new wage?
- New wage: $15 \times 1.10 = 16.50$ dollars per hour.
- Pay: $16.50 \times 30 = 495$.
- Estimate: $15 \times 30 = 450$, plus about 10% more, about 495 ✓.
Answer: \$495
Worked example: A leftover piece
A 10 ft board is cut into as many $1\frac{3}{4}$ ft pieces as possible. How much is left over?
- How many pieces fit: $$10 \div 1\frac{3}{4} = 10 \times \frac{4}{7} = \frac{40}{7} = 5\frac{5}{7}$$ so 5 whole pieces.
- They use $5 \times 1\frac{3}{4} = 8\frac{3}{4}$ ft.
- Left over: $10 - 8\frac{3}{4} = 1\frac{1}{4}$ ft. It is less than one piece ✓.
Answer: $1\frac{1}{4}$ ft