Percentages

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Every percentage needs a named base.

The central idea: name the base

A percent is a ratio per 100. To find p% of a quantity B, calculate $(p/100)B.$ Here B is the base: the amount representing 100%. The same numerical change can produce different percentages when the base changes.

Translate words into multipliers

Statement	Equation
A is p% of B.	$A = (p/100)B$
A is p% greater than B.	$A = (1 + p/100)B$
A is p% less than B.	$A = (1 - p/100)B$
A is k times B.	$A = kB$

“230% of” means a factor of 2.30, while “230% greater than” means a factor of 3.30. A positive quantity cannot fall by more than 100% and remain nonnegative, but an increase can exceed 100%.

Three questions, one equation

Start from $part = (p/100)(whole).$ A question can ask for any of its three ingredients. For an unknown percentage, divide the part by the whole and multiply by 100. For an unknown whole, divide the part by the decimal rate. Label which quantity represents 100% before solving.

Percent change versus a new value as a percent

$\frac{new − original}{original}$

For a decrease, the signed result is negative; its positive magnitude describes the percent decrease. A new value of 150 relative to an original 120 is 125% of the original but represents a 25% increase. These are not interchangeable answers.

Percentage points are not relative percentages

When a percentage changes from 30% to 36%, it rises by 6 percentage points. Relative to the original percentage, the rise is $6/30 = 20%.$ Use subtraction for percentage points and a change-over-original quotient for relative percent change. For a stated uncertainty of $\pm3$ percentage points around 40%, the endpoints are 37% and 43%, not 38.8% and 41.2%.

Reverse changes and successive changes

Reverse the operation, not the wording

After a 25% discount, the final price is $0.75P.$ Recover P by dividing the final price by 0.75. Adding 25% to the reduced price uses a different base and does not undo the discount.

Likewise, if A is 40% greater than B, then $A = 1.4B.$ The amount by which B is less than A is $0.4B,$ but its base is now $1.4B.$ The reverse comparison is $(0.4/1.4)(100)%,$ not 40%. A convenient original value such as 100 often makes this change of base visible.

Multiply factors when the base changes

For sequential changes, use $final = initial \times (factor 1) \times (factor 2) \dots.$

An increase of 10% followed by an increase of 20% has factor $1.10(1.20) = 1.32,$ a 32% increase. Do not add the percentages. Equal percent increases and decreases also do not cancel: $(1+r)(1-r) = 1-r 2$ for a rate r written as a decimal.

But some percentages deliberately share one base

If a tax and a tip are both calculated on the pre-tax meal price, their amounts can be added as percentages of that same base. If the tip instead applies to the tax-inclusive total, use successive multipliers. The wording, not a memorized shopping rule, determines the calculation.

A base ledger prevents most percentage errors

For each percentage, write “__________ percent of __________.” Identify whether the second blank is the original amount, a changed amount, a subgroup, or another percentage. Only then choose addition, multiplication, or division.

Repeated change and interest models

Repeated growth by rate r per equal interval gives $A = P(1 + r) t$ when t counts intervals. Simple interest is a different stated model: $I = Prt,$ with interest calculated only on the original principal, and balance $P + I.$ Do not assume one model from the word “interest” alone. This guide supplies the simple-interest formula in the relevant example; its role is interpreting the given relationship, not learning banking rules.

Percentages inside larger models

Nested groups

If 70% of a group are members and 40% of those members attend an event, attendees make up $0.70(0.40) = 0.28$ of the full group. The second fraction is applied to the first subgroup. A two-stage tree or a convenient total such as 1,000 can help, but is not essential.

Combined rates need weights

An overall success rate is total successes divided by total trials. If group sizes are $n 1,$ $n 2$ and their success proportions are $p 1,$ $p 2,$ then $\frac{n_{1} p_{1} + n_{2} p_{2}}{n_{1} + n_{2}}$

The unweighted average of $p 1$ and $p 2$ is correct only when the groups have equal sizes. Recover the counts first when the table makes them easy to calculate. An overall rate should fall between the two group rates for positive group sizes.

Percent concentration: track what is conserved

For a concentration by mass, $proportion = solute mass/total solution mass.$ If only water evaporates, the solute mass stays fixed while the denominator decreases. If water is added, the denominator increases. If two solutions are mixed, add solute masses and total masses separately. Never average concentrations without accounting for the sizes of the two solutions.

Products and unknown percent changes

Revenue is price times quantity. Its multiplier is the price multiplier times the quantity multiplier. If quantity falls but revenue rises, the price multiplier must more than compensate. The same reasoning applies to area as a product of two lengths or total production as rate times time: first identify the underlying mathematical relationship, then attach change factors to its inputs.

Answer the requested form

A proportion of 0.27, a percentage of 27%, and a count of 27 can represent different answers. If the question asks “what percent,” enter 27, not 0.27. If it asks for a probability, 0.27 may be correct. Write the interpretation before removing units for a numeric response.

What to practice next

Use Examples 2.01–2.06 for bases and language; 2.07–2.10 for successive and reverse changes; and 2.11–2.15 for nested groups, weighted rates, and multistep models.

15 worked examples

Try the question before reading the solution. The examples progress from Foundation to Challenge.

2.01. Translate a percent into a multiplication factor

A collection contains 250 photographs, and 18% are black-and-white. How many photographs are black-and-white?

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Recognize the structure. The whole is 250. Eighteen percent means 18 out of every 100 of that whole.

Work it through. Write $18% = 18/100 = 0.18.$ Then

$black-and-white photographs = 0.18(250) = 45.$

The result is a count rather than a percentage.

Answer: 45 photographs.

Check. 10% of 250 is 25 and 8% is 20, for a total of 45.

Avoid the trap. Multiplying by 18 instead of 0.18 makes the result 100 times too large. A proper subset cannot exceed all 250 photographs.

2.02. Find the percentage, not just the fraction

Of 120 invitations sent, 42 receive a reply. What percentage of the invitations receive a reply?

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Recognize the structure. The reference whole is the number sent, 120.

Work it through. The fraction receiving replies is $42/120 = 0.35.$ Convert the fraction to a percentage:

$\frac{42}{120}$

For a student-produced response asking for the percentage, the numerical entry is 35, not 0.35.

Answer: 35%.

Check. $0.35(120) = 42,$ recovering the given number of replies.

Avoid the trap. The decimal share 0.35 and the percentage 35% express the same proportion, but a question asking for the number of percent requires 35.

2.03. Recover a price before a discount

After a 30% discount, a jacket costs $84. What was its price before the discount?

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Recognize the structure. The sale price is 70% of the original price, not 30% of it.

Work it through. Let P be the original price. Then $(1 - 0.30)P = 84,$ so

$\frac{84}{0.70}$

Reversing a multiplication requires division by the same factor.

Answer: $120.

Check. 30% of $120 is $36, and $120 - 36 = 84.$

Avoid the trap. Increasing $84 by 30% does not undo a 30% discount, because the increase would use a different base.

2.04. Use the original value in a percent change

A club’s membership increases from 64 to 80. What is the percent increase?

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Recognize the structure. Percent change compares the change with the original value.

Work it through. The increase is $80 - 64 = 16.$ Relative to the original 64,

$\frac{16}{64}$

The growth factor is $80/64 = 1.25,$ which also shows a 25% increase.

Answer: 25%.

Check. One fourth of 64 is 16, and $64 + 16 = 80.$

Avoid the trap. Using 80 as the denominator gives 20%, which describes how much smaller 64 is than 80, not how much membership increased from 64.

2.05. Distinguish percent change from percentage points

The share of customers choosing electronic receipts rises from 18% to 24%. By how many percentage points does the share increase?
A. 6 B. 24
C. 33 $\frac{1}{3}$ D. 75

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Recognize the structure. Percentage points measure the arithmetic difference between two percentages.

Work it through. Subtract the percentages: $24 - 18 = 6$ percentage points. By contrast, the relative increase in the share is $\frac{24 − 18}{18}$

Both descriptions can be correct, but they answer different questions.

Answer: A: 6 percentage points.

Check. An 18% share plus 6 percentage points is 24%. A 6% relative increase would instead give $18%(1.06) = 19.08%.$

Avoid the trap. Do not append a percent sign to an answer in percentage points and assume the meanings are interchangeable. Identify the requested comparison first.

2.06. Keep tax and tip bases separate

A meal costs $50 before tax and tip. Sales tax is 8% of the meal price, and the tip is 18% of the pre-tax meal price. What is the total amount paid?

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Recognize the structure. The wording gives the same base, $50, for both added percentages.

Work it through. Tax is $0.08(50) = 4$ dollars, and tip is $0.18(50) = 9$ dollars. The total is

$50 + 4 + 9 = 63.$

Because the bases match, this can also be written $50(1 + 0.08 + 0.18) = 63.$

Answer: $63.

Check. The additions total 26% of $50, or $13. The final amount is $13 above the meal price.

Avoid the trap. 50(1.08)(1.18) would charge the tip on the after-tax amount. That is not the specified rule. Add rates only when they truly apply to the same unchanged base.

2.07. Combine successive changes by multiplying

A quantity increases by 20% and then decreases by 20% of its new value. What is the overall percent change from the original quantity?

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Recognize the structure. The percentages act on different bases. Use a multiplier for each stage.

Work it through. Starting with Q, the final quantity is

$Q(1.20)(0.80) = 0.96Q.$

The final amount is 96% of the original, so it is 4% less than the original.

Answer: A 4% decrease.

Check. Use an original value of 100: it becomes 120, then loses $0.20(120) = 24,$ ending at 96.

Avoid the trap. Equal increase and decrease percentages do not generally cancel. The second 20% is a percentage of 120 in the check, not of 100.

2.08. Interpret a percentage greater than 100

The equation A = 2.35B relates two positive quantities. Quantity A is what percent greater than quantity B?
A. 35% B. 135%
C. 235% D. 335%

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Recognize the structure. The multiplier includes the original 100% of B. “Greater than” asks only for the additional part.

Work it through. Subtract one whole copy of B:

$A - B = 2.35B - B = 1.35B.$

Thus the increase relative to B is $1.35B/B = 1.35 = 135%.$

Answer: B: 135%.

Check. When $B = 100,$ $A = 235.$ The excess is 135, which is 135% of the base 100.

Avoid the trap. 235% of B means 135% greater than B. The word “of” is not interchangeable with a statement of increase.

2.09. Reverse two discounts in the right direction

A store discounts an item by 20%, then applies an additional 10% discount to the reduced price. The final price is $144. What was the original price?

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Recognize the structure. The final price is the original multiplied by both remaining fractions.

Work it through. Let P be the original price. Then

$P(0.80)(0.90) = 144 0.72P = 144 P = 200.$

The combined discount is 28%, because 72% remains.

Answer: $200.

Check. The first discount takes $200 to $160; the second takes $160 to $144.

Avoid the trap. Adding 20% and 10% to get a 30% discount incorrectly gives $144/0.70.$ The second discount applies to a smaller base.

2.10. Reverse a percent comparison

The price of item A is 60% greater than the price of item B. The price of B is what percent less than the price of A?

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Recognize the structure. The reference whole changes from B in the first statement to A in the question.

Work it through. Let B cost 100 units. Then A costs 160 units. B is 60 units below A, so the requested percentage is

$\frac{160 − 100}{160}$

The difference is unchanged, but the denominator is now 160.

Answer: 37.5%.

Check. $160(1 - 0.375) = 160(0.625) = 100.$

Avoid the trap. A 60% increase does not correspond to a 60% decrease in the reverse direction. Name the base each time the comparison changes.

Another route. Algebraically, $A = 1.6B$ implies $B = A/1.6 = 0.625A,$ which is 37.5% below A.

2.11. Take a percentage of a subgroup

A school has 400 students. Of these students, 60% belong to a club. Of the club members, 25% volunteer at an event. How many students both belong to a club and volunteer at the event?

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Recognize the structure. The second percentage is conditional on belonging to the first group.

Work it through. There are $0.60(400) = 240$ club members. Of those, $0.25(240) = 60$ volunteer. In one expression, $400(0.60)(0.25) = 60.$

The combined share of all students is $0.60(0.25) = 0.15,$ or 15%.

Answer: 60 students.

Check. One quarter of 240 is 60. This count is below both the 240 club members and all 400 students.

Avoid the trap. Adding 60% and 25% has no meaningful interpretation here. Also, the problem gives no information about volunteers who are not club members.

2.12. Weight percentages by their group sizes

In a 50-person group, 80% complete a survey. In a separate 150-person group, 60% complete it. What percentage of all 200 people complete the survey?

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Recognize the structure. Combine counts first. The two group percentages have different denominators.

Work it through. The response counts are $0.80(50) = 40$ and $0.60(150) = 90.$ Therefore

$\frac{40 + 90}{50 + 150}$

The larger group has three times the weight of the smaller one.

Answer: 65%.

Check. The answer lies between 60% and 80%, closer to 60% because the larger group has the 60% response rate.

Avoid the trap. The unweighted average, 70%, would be correct only if the groups had equal sizes. Percentages are not counts.

2.13. Separate simple interest from repeated growth

In a hypothetical account, simple interest is calculated as I = Prt, where P is the initial deposit, r is the annual interest rate written as a decimal, and t is time in years. An initial deposit of $1,600 earns simple interest at 4.5% per year for 3 years. What is the total balance after 3 years?

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Recognize the structure. The given rule calculates interest on the unchanged initial deposit. The question asks for balance, not interest alone.

Work it through. Use $r = 0.045$ and $t = 3:$

$I = 1,600(0.045)(3) = 216.$

Add the initial deposit to obtain the balance: $1,600 + 216 = 1,816$ dollars.

Answer: $1,816.

Check. Each year adds $0.045(1,600) = 72$ dollars, so three years add $3(72) = 216$ dollars.

Avoid the trap. $1,600(1.045) 3$ is a compound-growth model, not the supplied simple-interest rule. The answer 216 also misses the request for the total balance.

2.14. Preserve the numerator when the whole changes

A 250-gram solution is 12% salt by mass. Exactly 50 grams of water evaporate, and no salt is lost. What percentage of the remaining solution’s mass is salt?

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Recognize the structure. The salt mass stays fixed while the total solution mass decreases.

Work it through. Initial salt mass is $0.12(250) = 30 g.$ After evaporation, the solution has mass $250-50 = 200 g.$ Therefore

$\frac{30}{200}$

The new percentage uses the new total mass.

Answer: 15%.

Check. The solution initially contained 220 g of water. Removing 50 g leaves 170 g of water plus 30 g of salt, totaling 200 g.

Avoid the trap. Subtracting 50 from the salt mass violates the statement that only water leaves. Keeping 250 as the denominator overlooks the changed whole.

2.15. Recover a hidden percentage from revenue

A business increases its price per item by p%. The number of items sold falls by 20%, but total revenue increases by 12%. Assuming revenue equals price per item times items sold, what is p?

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Recognize the structure. Revenue is a product. Its multiplier is the product of the price multiplier and the quantity multiplier.

Work it through. Let original price and quantity be P and Q. Then

$\frac{p}{100}$

Cancel the positive original revenue PQ and divide by 0.80:

$\frac{p}{100}$

Answer: 40.

Check. A price multiplier of 1.40 and quantity multiplier of 0.80 give $1.40(0.80) = 1.12,$ the stated revenue multiplier.

Avoid the trap. Adding 20 and 12 to get a 32% price increase ignores the fact that fewer items are sold at the higher price. Multiply the factors before interpreting the percentage.

Section 2: exit questions

Try the five section-exit questions before checking the explanations. Use scratch paper for your reasoning and enter the requested numerical value for a question without choices.

Topic practice