Percent Problems and Percent Change

Part, percent, whole

A percent is per hundred. In part=percent×whole, the part is the portion and the whole is the 100% reference. Convert percent to decimal. To find whole, divide part by the decimal rate.

Applied change

A discount is subtracted; tax, tip, and markup are added. Calculate the change, then answer whether the question requests change or final amount. Percent change is (new−original)/original×100%.

Percentage points

A percentage point is the direct difference between percentages. From 40% to 50% is 10 points but a 25% relative increase.

Scroll the table sideways if needed.

Worked examples

Worked example 1: 18 is what percent of 72? Divide 18/72=0.25, so 25%.

Worked example 2: A $64 item rises 12.5%. The increase is $8; final price is $72. Reporting $8 would answer only the change.

A reliable way to reason

First identify the reference whole. In “18 is 30% of what number,” 18 is the part and the unknown is the whole, so 18=0.30w and w=60. In “what is 30% of 18,” 18 is the whole, so the answer is 5.4. The same numbers produce different results because their roles change.

Successive percent changes use new reference amounts. A $100 item increased 20% becomes $120. A later 20% decrease is $24, leaving $96; equal percent increase and decrease do not cancel because the second percent uses 120 as its whole. A multiplier makes this visible: increase by r uses 1+r, while decrease by r uses 1−r. For tax or tip, decide whether the question asks for the added amount or total. For comparisons of rates, name the measure: from 30% to 36% is 6 percentage points, while the relative increase is 6/30=20%. Including the word “points” prevents these answers from being confused.

Guided reasoning and error checks

For a complete applied solution, label three lines: reference, change, and final. Suppose a $90 price decreases 15%. The reference is the original $90. The change is 0.15×90=$13.50. The final price is $90−$13.50=$76.50. Check by using the remaining multiplier: 1−0.15=0.85, and 0.85×90=$76.50. Both paths agree.

Now suppose the final price after a 15% discount is $76.50 and the original is unknown. Do not subtract 15% from the final price; write 0.85w=76.50 and divide to obtain $90. The whole changes with the wording. For an increase from 80 to 92, the change is 12 and the original reference is 80, so 12/80=15%. Dividing by 92 would answer a different comparison. State “15% increase” to show both direction and reference.

Final self-check

A percent equation should also identify its unit. In 0.18×250 people=45 people, the decimal rate has no unit, so the product keeps the people unit. In $48÷0.60=$80, dollars divided by a unitless rate remains dollars. Tracking the unit helps distinguish a whole amount from a rate or percentage-point difference. Write the requested unit beside the final answer.

Your quick summary

• Convert percent to decimal before multiplying.
• Separate change from final amount.
• Use original as the percent-change denominator.
• Percentage points subtract two rates directly; relative percent change divides the change by the original rate.