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Key terms
Percent base
The whole quantity to which a percent applies.
Multiplier
A growth or decay factor such as 1.12 or 0.84.
Percentage point
The arithmetic difference between two percentages.
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Step 2 of 7·Try it firstDone
Try it first
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Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.
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Step 3 of 7·LearnDone
The big idea
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See the structure
A percent is a rate per 100. Translate “p percent of B” as (p/100)B and identify B, the base, before calculating.
Key idea
Name quantities before calculating
Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.
Build a reliable method
A percent increase multiplies by 1+r; a decrease multiplies by 1−r. Successive changes multiply their factors and usually do not cancel.
Explore in Desmos
Apply successive percent factors
A $260 price rises 10% and then falls 20%. Read $286 after the increase and $228.80 after the discount. Compare the third line to see why combining the labels as a single -10% change gives the wrong base.
The calculator is live: change anything and see what happens.
Apply successive percent factors
Degree mode
Press Escape to leave the calculator (twice if a menu is open).
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Stretch to a 36
Percentage points compare percentages by subtraction. Percent change divides that difference by the original percentage.
Trap
A believable answer to the wrong question
The choices often include a correct intermediate number. Before selecting it, reread the highlighted ask and check the unit, sign, and domain.
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Step 4 of 7·Worked examplesDone
Worked examples
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Skip check · 2 questions
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Worked example
Example 1: Build, solve, check
Problem
A subscription price of $260 increases 10% and then receives a 20% loyalty discount. What is ==the final price, in dollars==?
Turn each percent change into a multiplier and apply the multipliers in chronological order.
For these numbers, the calculation is 260⋅1.1⋅0.8.
Which percent relationship uses the correct base?
Evaluate to get 228.8.
Check the result against the original quantities.
Which calculation verifies the percent result from the original amount?
Worked example
Example 2: Work backward from a sale price
Problem
After a 15% discount, a bicycle costs $306. What was the original price?
A 15% discount leaves 85% of the original, so 306=0.85p.
To isolate p, divide by 0.85.
Which percent relationship uses the correct base?
p=360, so the original price was $360.
Check forward: 15% of $360 is $54, and 360−54=306.
Which calculation verifies the percent result from the original amount?
Faded example · you finish it
Example 3: Points or percent change?
Problem
When approval rises from 48% to 60%, what are the percentage-point increase and the percent increase relative to 48%?
The percentage-point increase comes from subtraction: 60−48=12 points.
Which expression gives the relative percent increase?
The relative percent increase is
Desmos way
Forward and reverse percent calculations
Type each full expression on its own line. For reverse percent, divide by the remaining factor 0.85.
Read 228.8, 360, and 0.25. The last decimal is 25%, while the raw percentage-point change is 12.
Forward and reverse percent calculations
Degree mode
Press Escape to leave the calculator (twice if a menu is open).
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Step 5 of 7·PracticeDone
Practice
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Step 6 of 7·Spot the mistakeDone
Spot the mistake
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Spot the mistake
Where did it go wrong?
One line has an error. Click the line where it first goes wrong.
What went wrongSuccessive percents use different bases: 1.20(0.80)=0.96.
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Step 7 of 7·Exit ticketDone
Exit ticket
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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36
These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.