Percents as ratios comparing a number to one hundred, the three notations, the percent verbal model and its algebraic form, the three kinds of percent question distinguished by which letter is unknown, converting a percent to a decimal before substituting, and identifying the base number in a real problem.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 3 — Solving Linear Equations
Percents
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 183-188 — the lesson these objectives are drawn from
Warm-up
Lesson 3.8 defined a ratio. A percent is a ratio with its denominator fixed.
Discussion prompt
Write forty percent as a fraction with denominator 100, then as a decimal. What does the word percent literally tell you to do?
Hint: Split the word into two parts.
Answer:
\[ 40\% = \tfrac{40}{100} = 0.40 \]
Per cent means per hundred, so the word itself names the denominator. Every percent is a ratio comparing a number to one hundred, which is why converting to a decimal is always a division by a hundred and never anything else.
Concept
A percent is a ratio comparing a number to one hundred. Every percent question is the same equation — the number compared to the base equals the percent times the base number — with a different one of the three quantities missing.
base number — The number that is being compared to in a percent equation. In thirty percent of seventy feet, the base number is seventy feet.
\[ a = p \cdot b \]
Which of a, p and b is unknown is decided by the wording, and once it is known the solving is ordinary.
Figure (svg): The percent verbal model with its three labelled parts and the algebraic model beneath
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 183-183
Section
Section 1
Concept
A percent compares a number to one hundred. It can be written as a fraction over a hundred, as a decimal, or as a number followed by a percent sign, and all three mean the same thing.
Converting to a decimal always means dividing by a hundred, which moves the decimal point two places left.
Figure (svg): Forty percent written three ways: as a fraction over one hundred, as a decimal, and with a percent sign
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 183-183 — the definition of a percent and its three forms
Picture it
Three notations for one quantity.
Figure (svg): Forty percent written three ways: as a fraction over one hundred, as a decimal, and with a percent sign
The decimal is the form that goes into the equation, which is why the conversion is the first step of every percent problem.
Worked example
Moving in both directions, since questions use all three.
\[ \text{Write } 30\%, \; 25\%, \; 500\% \text{ as fractions and as decimals.} \]
Write each as a fraction over one hundred
Why: The number in front of the percent sign is the numerator.
\[ \frac{30}{100}, \frac{25}{100}, \frac{500}{100} \]
Simplify where useful
Why: Twenty-five over a hundred is one quarter; five hundred over a hundred is five.
\[ \frac{3}{10}, \frac{1}{4}, 5 \]
Convert each to a decimal by dividing by a hundred
Why: Move the decimal point two places left.
\[ 0.30, 0.25, 5.00 \]
Note the one above a hundred
Why: Five hundred percent is five, a number greater than one — perfectly ordinary.
\[ 500 \% = 5 \]
Figure (svg): The solution to Worked example convert between the three forms shown as a ladder of expressions, one row per algebraic move
\[ 30\% = 0.30, \quad 25\% = 0.25, \quad 500\% = 5 \]
Verify: convert each decimal back
Why: Multiplying 0.30 by a hundred returns thirty, and multiplying 5 by a hundred returns five hundred. Every conversion reverses cleanly, since dividing and multiplying by a hundred are inverse operations.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 183-183
Matching
Every conversion is a division by one hundred.
Match the pairs
Why: Each decimal is its percent divided by a hundred, which moves the decimal point two places left. The third is the one most often got wrong, since 7.5 percent looks like it should give 0.75 rather than 0.075, and the fourth shows that a percent above a hundred gives a decimal above one.
Worked example
The conversion rule does not change when the percent has a decimal in it.
\[ \text{Write } 7.5\% \text{ and } 0.4\% \text{ as decimals.} \]
Divide 7.5 by a hundred
Why: Move the point two places left.
\[ 0.075 \]
Divide 0.4 by a hundred
Why: Two places left again, adding zeros as needed.
\[ 0.004 \]
Sanity-check both
Why: Both percents are small, so both decimals should be well below one.
Note the common slip
Why: Writing 7.5 percent as 0.75 is the commonest error here, and it is off by a factor of ten.
\[ \text{not } 0.75 \]
Figure (svg): The solution to Worked example percents that are not whole numbers shown as a ladder of expressions, one row per algebraic move
\[ 7.5\% = 0.075, \quad 0.4\% = 0.004 \]
Verify: compare with a familiar percent
Why: Ten percent is 0.10, so 7.5 percent must be a little less — and 0.075 is, while 0.75 would be seventy-five percent. Anchoring against a percent you know is the fastest way to catch a misplaced decimal point.
Trap
\[ \text{What is } 30\% \text{ of } 70? \]
Substitute 30 for the percent
Why: Thirty is the number written, so it looks like the value to use.
\[ a = 30 \cdot 70 = 2100 \]
Two thousand one hundred is thirty times the whole, not thirty percent of it. The percent sign was an instruction to divide by a hundred and it was ignored.
\[ 30\% = 0.30 \;\Longrightarrow\; a = 0.30 \cdot 70 = 21 \]
Convert the percent to a decimal or a fraction before substituting
Why: The percent sign is not decoration; it means the number is a count of hundredths.
A size check settles it instantly: a percent below a hundred must give an answer smaller than the base, and 2100 is thirty times larger.
Sorting
Compare each percent with one hundred.
Sort into buckets
Sort each percent by what it does to the base number.
Predicting which side of the base the answer falls on is the fastest check available, and it costs no arithmetic at all.
Faded example
Divide the percent by a hundred first.
Fill in the blanks
30\% = 0.30 \;\Longrightarrow\; a = 0.30 \cdot 70 = 21
Why: Thirty percent is thirty hundredths, which is 0.30, and multiplying that by seventy gives twenty-one. The two blanks are the same number deliberately: the conversion happens once, before the substitution, and never during it.
Socratic
The rule is easy to remember and worth understanding.
Discussion prompt
Explain why converting a percent to a decimal moves the point two places to the left, using the meaning of the word percent. Then say what would happen if the word had meant per thousand instead.
Hint: The word names a denominator.
Answer:
Percent means per hundred, so thirty percent is thirty hundredths — the fraction thirty over one hundred. Dividing by a hundred moves every digit two place-values to the right, which is the same as moving the decimal point two places left.
Per thousand would divide by a thousand and move the point three places. That unit exists and is written with a per-mille sign, used in some financial and scientific contexts. The number of places is simply the number of zeros in the denominator the word names.
Section
Section 2
Concept
The percent verbal model says the number being compared to the base equals the percent times the base number. Labelling the three quantities turns any percent question into a one-step equation.
\[ a = p \cdot b \]
The percent has no units of its own, which is why the answer carries the base's units.
| Quantity | Letter | Units |
|---|---|---|
| Number compared to base | a | same as b |
| Percent | p | none |
| Base number | b | assigned by the problem |
Figure (svg): The percent verbal model with its three labelled parts and the algebraic model beneath
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 183-183 — the percent verbal model and its labels
Picture it
Each phrase sits directly above the letter it became.
Figure (svg): The percent verbal model with its three labelled parts and the algebraic model beneath
This is the same layout as Lesson 1.6's models, and it works the same way: a wrong model can be found by reading rather than by solving.
Worked example
This is Example 1 from the textbook. The number compared to the base is unknown.
\[ \text{Find } a \text{ in } \; a = (0.30)(70). \]
Write the verbal model
Why: The number compared to the base equals the percent times the base number.
\[ a = p \cdot b \]
Assign labels with units
Why: a is in feet, p is 30 percent which is 0.30 with no units, b is 70 feet.
Write and evaluate the algebraic model
Why: Nought point three times seventy.
\[ a = 21 \]
Attach the unit
Why: The percent has no units, so the answer carries the base's unit of feet.
\[ 21\text{ feet} \]
Figure (svg): A bar showing 70 feet with 30 percent of it shaded
\[ a = (0.30)(70) = 21 \text{ feet} \]
Verify: check the size against the base
Why: Thirty percent is less than half, so the answer should be less than thirty-five — and twenty-one is. A rough check like this catches the un-converted percent immediately, since that would have given 2100.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 183-183
Sorting
In each phrase, identify the base number.
Sort into buckets
Sort each item by which role it plays in its own question.
The word of is the reliable marker: whatever follows it is the base. Finding the base first makes the other two roles fall out automatically.
Worked example
Example 2 from the textbook. This time the base number is unknown.
\[ \text{Solve } \; 14 = (0.25)b. \]
Identify which quantity is unknown
Why: The question asks what amount, so the base number is missing.
Convert the percent
Why: Twenty-five percent is one quarter, or 0.25.
\[ p = \frac{1}{4} \]
Write the algebraic model
Why: Fourteen equals one quarter of b.
\[ 14 = (\frac{1}{4}) b \]
Solve by multiplying by the reciprocal
Why: Multiply both sides by four.
\[ b = 56 \]
Figure (svg): The solution to Worked example fourteen dollars is 25 percent of what shown as a ladder of expressions, one row per algebraic move
\[ 14 = \tfrac{1}{4}b \;\Longrightarrow\; b = 56 \text{ dollars} \]
Verify: check by computing the percent of the answer
Why: A quarter of fifty-six is fourteen, which is the amount given. And the base is larger than the compared number, as it must be whenever the percent is below a hundred.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 184-184
Error analysis
The student set up three percent equations. Two are wrong.
Annotate
On: \( \begin{aligned} \text{30\% of 70} &: \; a = 30 \cdot 70 \\ \text{14 is 25\% of what} &: \; 14 = 0.25b \\ \text{20\% of 60} &: \; a = 0.20 \cdot 20 \end{aligned} \)
Both errors are caught by a size check. Thirty percent of seventy must be under seventy, and twenty percent of sixty must be under sixty — neither wrong answer satisfies its check.
Faded example
Assign the three quantities before solving.
Fill in the blanks
\text0.30 30\% \text70 70? \;\rightarrow\; a = ___ \cdot ___
Why: The percent converts to 0.30 and the base is the seventy named after the word of, giving a equal to twenty-one. Setting the equation up with both quantities identified is what makes the solving a single multiplication.
Elimination
The question is: fourteen dollars is 25 percent of what amount?
Eliminate the wrong options
Which equation is correct?
Survives elimination: A
Why: The fourteen is the number compared to the base, the base is unknown, and the percent converts to 0.25. Solving gives fifty-six. Option B is the tempting one: it treats the part as the whole, and its answer of 3.50 is smaller than fourteen, which a size check rejects immediately.
Socratic
The other two quantities in the model both have units.
Discussion prompt
Explain why the percent in the model has no units of its own, using the fact that a percent is a ratio. Then say what that means for the units of the answer.
Hint: A ratio compares two quantities in the same unit.
Answer:
A percent compares a number to one hundred, and both are pure counts of the same kind of thing, so the units cancel exactly as they did in Lesson 3.8's ratios. Thirty percent is thirty hundredths of whatever it is applied to, and the thirty and the hundred carry no unit between them.
It means the answer carries the base's units unchanged. Thirty percent of seventy feet is twenty-one feet, and thirty percent of seventy dollars is twenty-one dollars — the percent multiplies the number and leaves the unit alone. That is why the labels table assigns units to a and b but not to p.
Section
Section 3
Concept
There are three basic types of percent problem, distinguished by which of the three quantities is unknown. The equation is the same in every case, and the wording tells you which letter to solve for.
Figure (svg): The three kinds of percent question, each with its unknown marked
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 183-184 — the three basic types of percent problem
Picture it
Each row is the same equation with a different unknown.
Figure (svg): The three kinds of percent question, each with its unknown marked
Noticing which letter is missing is the whole of the setup. After that, each is a one-step equation of the kind Lesson 3.2 handled.
Worked example
This is Example 3 from the textbook. One hundred and thirty-five is what percent of twenty-seven?
\[ \text{Solve } \; 135 = p(27) \; \text{ for } p. \]
Identify the unknown
Why: The question asks what percent, so p is missing.
Write the algebraic model
Why: One hundred and thirty-five equals p times twenty-seven.
\[ 135 = 27 p \]
Divide both sides by 27
Why: One hundred and thirty-five over twenty-seven is five.
\[ p = 5 \]
Convert back to a percent
Why: Multiply by a hundred, since p was a decimal.
\[ 500 \% \]
Figure (svg): The solution to Worked example the percent is unknown shown as a ladder of expressions, one row per algebraic move
\[ 135 = 27p \;\Longrightarrow\; p = 5 = 500\% \]
Verify: check that a percent above a hundred makes sense here
Why: One hundred and thirty-five is five times twenty-seven, so the compared number exceeds the base — and a percent above a hundred is exactly what that means. Five hundred percent is not an error; it says the first number is five times the second.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 184-184
Discrimination
Read the wording to find the missing quantity.
Sort into buckets
Sort each question by which quantity it asks for.
Worked example
Guided Practice 1 to 4. One of each type, and one with a percent above a hundred.
\[ \text{What is } 15\% \text{ of } 100 \text{ m? } 12 \text{ is } 60\% \text{ of what? } 8 \text{ is what percent of } 20? \; 20 \text{ is what percent of } 8? \]
Solve the first: a is unknown
Why: Nought point one five times a hundred.
\[ 15\text{ metres} \]
Solve the second: b is unknown
Why: Twelve equals 0.60 b, so divide by 0.60.
\[ b = 20 \]
Solve the third: p is unknown
Why: Eight equals p times twenty, so p is 0.4.
\[ 40 \% \]
Solve the fourth: p is unknown again
Why: Twenty equals p times eight, so p is 2.5.
\[ 250 \% \]
Figure (svg): The solution to Worked example four from guided practice shown as a ladder of expressions, one row per algebraic move
\[ 15 \text{ m}, \quad 20, \quad 40\%, \quad 250\% \]
Verify: compare the last two
Why: Eight is forty percent of twenty, and twenty is two hundred and fifty percent of eight. The same pair of numbers gives two very different percents depending on which is the base, which is exactly why identifying the base matters more than any arithmetic here.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 184-184
Trap
\[ \text{8 is what percent of 20?} \]
Divide the larger by the smaller, since percents are usually below a hundred
Why: Twenty over eight gives a tidier number, and answers below a hundred feel more normal.
\[ p = \tfrac{20}{8} = 2.5 = 250\% \quad \text{(wrong question answered)} \]
That answers the reverse question, twenty is what percent of eight. Eight is forty percent of twenty, and the two answers are not even close.
\[ 8 = p(20) \;\Longrightarrow\; p = \tfrac{8}{20} = 0.4 = 40\% \]
Put the quantity after the word of underneath, whatever its size
Why: The base is named by the wording, not chosen for convenience.
A size check confirms it: eight is less than twenty, so the percent must be below a hundred. Any answer above a hundred here has the base and the compared number the wrong way round.
Elimination
Someone computes 20 divided by 8 and gets 2.5, or 250 percent.
Eliminate the wrong options
Which question has that answer?
Survives elimination: A
Why: Dividing twenty by eight makes eight the base, which matches the wording twenty is what percent of eight. The four options use the same two numbers in four different roles and produce four different answers, which is why identifying the base is the whole of the setup.
Prediction
Compare the two numbers before computing.
Predict first
Is 135 more or less than 100 percent of 27?
Correct: More, since 135 is larger than 27.
\[ \tfrac{135}{27} = 5 = 500\% \]
Why: A hundred percent of the base is the base itself, so a compared number larger than the base means a percent above a hundred. One hundred and thirty-five is five times twenty-seven, giving five hundred percent. Comparing the two numbers first predicts the answer's range and catches a swapped base immediately.
Faded example
Divide the compared number by the base.
Fill in the blanks
8 = p(20) \;\rightarrow\; p = \tfrac200.4} = ___ = 40\%
Why: The base is twenty, so dividing eight by twenty gives 0.4, which is forty percent. The compared number is smaller than the base, so the percent had to come out below a hundred — a prediction worth making before the division.
Section
Section 4
Concept
Identifying the base is the decisive step in a real percent problem. In a discount the base is the original price; in an increase it is the value before the increase.
Using the sale price as the base instead of the original is the commonest error in discount problems.
Figure (svg): A price reduced from 40 to 30, with the discount percent computed from the reduction
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 186-186 — Example 4, finding the discount percent on a sale item
Picture it
The reduction is compared with the original price.
Figure (svg): A price reduced from 40 to 30, with the discount percent computed from the reduction
Ten dollars off forty is twenty-five percent. Ten off the sale price of thirty would be about thirty-three percent, which answers a question nobody asked.
Worked example
Example 4 in spirit. An item marked 40 dollars is on sale for 30 dollars.
\[ \text{Find the discount percent.} \]
Work out the reduction
Why: Forty minus thirty is ten dollars off.
\[ 10\text{ dollars off} \]
Identify the base
Why: A discount is compared with the original price, so the base is forty.
\[ b = 40 \]
Write the equation
Why: Ten equals p times forty.
\[ 10 = 40 p \]
Solve and convert
Why: Ten over forty is 0.25, which is twenty-five percent.
\[ 25 \% \]
Figure (svg): A price reduced from 40 to 30, with the discount percent computed from the reduction
\[ 10 = 40p \;\Longrightarrow\; p = 0.25 = 25\% \]
Verify: check by applying the discount
Why: Twenty-five percent of forty is ten, and forty minus ten is thirty — the sale price given. Working forwards from the answer to recover a figure the problem stated is the strongest available check.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 186-186
Sorting
The base is what the change is measured against.
Sort into buckets
Sort each situation by which quantity is the base.
Both columns follow the same rule: the base is whatever the other quantity is being compared to. Changes compare to the starting value and parts compare to the whole.
Worked example
The same reasoning with the base before the change rather than before the reduction.
\[ \text{A price rises from } 25 \text{ to } 30 \text{ dollars. Find the percent increase.} \]
Work out the change
Why: Thirty minus twenty-five is five dollars.
\[ 5\text{ dollars up} \]
Identify the base
Why: A percent increase is compared with the value before the increase.
\[ b = 25 \]
Write and solve the equation
Why: Five equals p times twenty-five, so p is 0.2.
\[ p = 0.2 \]
Convert and state
Why: Twenty percent increase.
\[ 20 \% \]
Figure (svg): The solution to Worked example an increase rather than a discount shown as a ladder of expressions, one row per algebraic move
\[ 5 = 25p \;\Longrightarrow\; p = 0.2 = 20\% \]
Verify: check against the wrong base
Why: Using thirty as the base would give five over thirty, which is about 16.7 percent — a different answer to a different question. The rise is measured against where it started, which is twenty-five.
Trap
\[ \text{40 dollars reduced to 30: } p = \tfrac{10}{30} \approx 33\% \]
Divide the reduction by the sale price
Why: The sale price is the number you end up paying, so it feels like the reference point.
Thirty-three percent off forty would be about 13.30, giving a sale price of about 26.70 rather than thirty. The answer does not reproduce the situation.
\[ p = \tfrac{10}{40} = 25\% \]
Compare the change with the value it started from
Why: A discount is a fraction of the original price, which is what the shopper was originally being asked to pay.
Always check by applying your percent to the base and confirming you recover the other figure the problem gave.
Elimination
An item marked 40 dollars sells for 30 dollars.
Eliminate the wrong options
Which calculation gives the discount percent?
Survives elimination: A
Why: The reduction is ten dollars and the base is the original forty, giving twenty-five percent. Option C is worth noticing: seventy-five percent is a correct and useful figure — the fraction of the price still paid — but it answers a different question from the one asked.
Missing information
A question can be perfectly well written and still be unanswerable.
Discussion prompt
An item is on sale for 30 dollars. What is the discount percent? Say exactly what is missing, and give two different answers depending on how the gap is filled.
Hint: A discount is measured against something.
Answer:
The original price is missing, so there is no base to compare against and no reduction to compute.
\[ \text{original } 40: \; p = \tfrac{10}{40} = 25\% \qquad \text{original } 50: \; p = \tfrac{20}{50} = 40\% \]
The same sale price gives very different discounts depending on where it started from, which is precisely why advertisements always state the original price alongside the sale one — and why a sale price alone tells a shopper nothing about the size of the saving.
Socratic
For a change, one of the two values has to be chosen.
Discussion prompt
Explain why a percent increase is measured against the value before the increase rather than after it. Then work out both versions for a rise from 25 to 30 and say why the two answers differ.
Hint: Ask what a percent increase is supposed to tell somebody.
Answer:
\[ \text{against } 25: \; \tfrac{5}{25} = 20\% \qquad \text{against } 30: \; \tfrac{5}{30} \approx 16.7\% \]
A percent increase is supposed to say how much bigger the new value is relative to where it started, so the starting value is the natural reference point. Measuring against the new value answers a different question — what fraction of the new total the increase represents.
The two differ because the denominators differ, and the gap grows with the size of the change. For a doubling, the increase is 100 percent of the old value but only 50 percent of the new one, and quoting the second figure would systematically understate every rise.
Section
Section 5
Concept
A common real question gives the value after a percent change and asks for the value before it. The base is still the original, so the equation compares the final value with the original rather than with the change.
\[ \text{final} = (1 + p)\cdot\text{original} \quad \text{or} \quad (1 - p)\cdot\text{original} \]
Subtracting the percent from the final value is a different and usually wrong calculation.
Figure (svg): Two columns contrasting a percent below one hundred with one above it
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 186-187
Picture it
Both are ordinary, and one is often mistaken for an error.
Figure (svg): Two columns contrasting a percent below one hundred with one above it
A price after a twenty percent rise is a hundred and twenty percent of what it was. Expressing the whole change as a single multiplier is what makes reversing it a single division.
Worked example
The sale price is given and the original is wanted.
\[ \text{After a } 25\% \text{ discount an item costs } 30 \text{ dollars. Find the original price.} \]
Express the sale price as a percent of the original
Why: Twenty-five percent off leaves seventy-five percent.
\[ \text{sale } = 0.75 \cdot\text{ original} \]
Write the equation
Why: Thirty equals 0.75 times the original.
\[ 30 = 0.75 b \]
Solve by dividing
Why: Thirty over 0.75 is forty.
\[ b = 40 \]
State the answer
Why: The original price was forty dollars.
\[ 40\text{ dollars} \]
Figure (svg): The solution to Worked example find the price before a discount shown as a ladder of expressions, one row per algebraic move
\[ 30 = 0.75b \;\Longrightarrow\; b = 40 \text{ dollars} \]
Verify: apply the discount to the answer
Why: Twenty-five percent of forty is ten, and forty minus ten is thirty — the sale price given. Working forwards recovers the stated figure, which confirms both the setup and the arithmetic.
Prediction
The same percent applied both ways is not symmetric.
Predict first
A price of 100 rises by 20 percent and then falls by 20 percent. What is it now?
Correct: 96.
\[ 100 \cdot 1.20 = 120 \qquad 120 \cdot 0.80 = 96 \]
Why: The rise takes it to 120, and a twenty percent fall from 120 is twenty-four, leaving ninety-six. The two percents have different bases — a hundred and then a hundred and twenty — so they do not cancel. This asymmetry is the single most useful fact about percent changes and catches out almost everybody the first time.
Worked example
A tip or a tax is added to a bill, and the total is more than a hundred percent of it.
\[ \text{A } 60 \text{ dollar bill has a } 15\% \text{ tip added. Find the total.} \]
Compute the tip
Why: Fifteen percent of sixty is 0.15 times 60.
\[ 9\text{ dollars} \]
Add it to the bill
Why: Sixty plus nine is sixty-nine.
\[ 69\text{ dollars} \]
Check the shortcut
Why: One hundred and fifteen percent of sixty is 1.15 times 60, which is also sixty-nine.
\[ 1.15 \cdot 60 = 69 \]
Note why the shortcut works
Why: Adding fifteen percent means keeping a hundred percent and adding fifteen more.
\[ 115 \% \]
Figure (svg): The solution to Worked example adding a percent rather than a percent of shown as a ladder of expressions, one row per algebraic move
\[ 60 + 0.15(60) = 1.15(60) = 69 \text{ dollars} \]
Verify: check the two routes agree
Why: Computing the tip separately and using the 1.15 multiplier both give sixty-nine. The single multiplier is the distributive property from Lesson 2.6 applied backwards, since 60 plus 0.15 times 60 is the quantity 1 plus 0.15, all times 60.
Trap
\[ \text{30 dollars after a 25\% discount} \]
Add 25 percent of the sale price to get back to the original
Why: The discount was twenty-five percent, so adding twenty-five percent should undo it.
\[ 30 + 0.25(30) = 37.50 \quad \text{(wrong)} \]
The discount was twenty-five percent of forty, not of thirty. Adding a percent of the smaller number cannot recover the larger one.
\[ 30 = 0.75b \;\Longrightarrow\; b = 40 \]
Express the sale price as a percent of the original and divide
Why: The base of the discount was the original, so the equation must compare against the original.
Percent changes do not reverse by applying the same percent the other way. A twenty-five percent fall is undone by a thirty-three percent rise, not by another twenty-five.
Elimination
After a 20 percent discount an item costs 48 dollars.
Eliminate the wrong options
Which calculation gives the original price?
Survives elimination: A
Why: A twenty percent discount leaves eighty percent, so the sale price is 0.80 times the original and dividing recovers sixty. Checking forwards confirms it: twenty percent of sixty is twelve, and sixty minus twelve is forty-eight.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Question | The base is | The calculation |
|---|---|---|
| 30% of 70 | 70, the whole | multiply by 0.30 |
| discount from 40 to 30 | 40, the original price | divide 10 by 40 |
| price after a 15% tip on 60 | 60, the bill | multiply by 1.15 |
All three use the same equation with different letters known. Identifying the base is what distinguishes them, and it is decided by the wording rather than by the numbers.
Socratic
A twenty percent rise followed by a twenty percent fall loses money.
Discussion prompt
Explain why the two twenty percents do not cancel, referring to their bases. Then work out what percent fall would exactly undo a twenty percent rise.
Hint: Compare the base of each change.
Answer:
The rise is twenty percent of a hundred, which is twenty, and the fall is twenty percent of a hundred and twenty, which is twenty-four. The second percent is applied to a larger base, so it removes more than the first added, and the net effect is a loss of four.
\[ 120x = 100 \;\Longrightarrow\; x = \tfrac{5}{6} \approx 0.833, \text{ a fall of about } 16.7\% \]
To return exactly to a hundred you need a fall of about 16.7 percent, since the fall is measured against the larger figure. This is why percent changes are usually converted to multipliers — 1.20 and its reciprocal — before being combined at all.
Comparison
Fill the blanks from memory before you scroll back. The equation is the same in every row.
Comparison matrix
| Question shape | Unknown | How to solve |
|---|---|---|
| What is 30% of 70? | a | multiply the base by the decimal |
| 14 is 25% of what? | b | divide by the decimal |
| 135 is what percent of 27? | p | divide by the base, then multiply by 100 |
Recognising the shape of the question is the whole setup. Each type is a one-step equation once the three quantities have been identified.
Pattern
Whether the question is abstract or a real discount, the same five moves cover it.
Step one is the decisive one. Two of the three quantities are usually obvious and the base is the one people choose wrongly, especially in discounts.
OpenStax Elementary Algebra 2e, §3.2 Solve Percent Applications §3.2
Check
The compared number is unknown. Convert first.
Check your understanding
What is 45 percent of 80 metres?
Answer: A
Why: Forty-five percent is 0.45, and 0.45 times 80 is 36 metres. The answer is less than half the base, which is right since forty-five percent is just under half.
Check
The base is unknown. Divide by the decimal.
Check your understanding
Eighteen is 40 percent of what number?
Answer: A
Why: The equation is 18 equals 0.40 b, so b is 18 over 0.40, which is 45. Checking forwards confirms it: forty percent of forty-five is eighteen. The base is larger than the compared number, as a percent below a hundred requires.
Check
A discount. Identify the base before dividing.
Check your understanding
A coat marked 80 dollars sells for 60 dollars. What is the discount percent?
Answer: A
Why: The reduction is twenty dollars and the base is the original eighty, so the discount is 20 over 80, which is 0.25 or twenty-five percent. Checking forwards: twenty-five percent of eighty is twenty, and eighty minus twenty is sixty.
Real world
A shop advertises 30 percent off, and a separate voucher gives a further 20 percent off at the till. An item is originally 200 dollars.
Discussion prompt
Compute the final price, then find the single percent discount that would have the same effect. Explain why it is not 50 percent, and say which order the two discounts should be applied in to get the lowest price.
Hint: Convert each discount to a multiplier and combine them.
Answer:
\[ 200 \cdot 0.70 = 140 \qquad 140 \cdot 0.80 = 112 \]
\[ \tfrac{112}{200} = 0.56, \text{ so } 44\% \text{ off in total} \]
The combined discount is forty-four percent, not fifty, because the second twenty percent is taken off the already-reduced 140 rather than off the original 200. Percent discounts multiply rather than add.
The order makes no difference: 200 times 0.80 times 0.70 also gives 112, since multiplication is commutative. That is worth knowing, because shops sometimes imply the order matters and it does not — though which discounts are allowed to combine at all is a separate question.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Can a percent be greater than 100?
Correct: Yes, whenever the compared number exceeds the base.
\[ 135 = p(27) \;\Longrightarrow\; p = 5 = 500\% \]
\[ \text{but } 8 = p(20) \;\Longrightarrow\; p = 0.4 = 40\% \]
Why: One hundred and thirty-five is five hundred percent of twenty-seven, since it is five times as large. A hundred percent means equal to the base, so anything larger than the base gives a percent above a hundred. This is entirely ordinary — a population that triples has grown by two hundred percent — and the belief that a hundred is a ceiling comes from percents most often being used for parts of a whole, where it genuinely is.
Explain it
They can compute percentages with a calculator and have never met the percent equation.
Discussion prompt
In no more than four sentences, explain how the three different-looking percent questions are really one equation. Then tell them the one word to look for that identifies the base, and why getting the base wrong matters more than any arithmetic slip.
Hint: The word is very short.
Answer:
A usable answer: every percent question says that one number equals a percent of another. That is a equals p times b, and the three types just differ in which of the three you are missing. Find the two you have, put them in, and solve for the third.
The word to look for is of — whatever follows it is the base. Getting the base wrong changes which question you are answering rather than merely giving a slightly wrong number: eight is forty percent of twenty, but twenty is two hundred and fifty percent of eight, and no amount of careful arithmetic recovers from choosing the wrong one.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: Converting is fixed by making the division by a hundred a separate written line. Deciding the unknown is fixed by learning the three question shapes — what is, of what, and what percent. The base is fixed by looking for the word of, or for the value before a change. Percents above a hundred are fixed by remembering that a hundred percent means equal to the base, so more than the base means more than a hundred. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page write the percent verbal model with the algebraic model directly underneath, each phrase above the letter it became. Below that, write the three kinds of question as three rows, each with an example, the letter that is unknown, and the operation that solves it. In the middle, work one discount problem in full, marking clearly which quantity you chose as the base and why, and check it by applying your percent forwards. Near the bottom, take a price, raise it by twenty percent and then lower the result by twenty percent, and write down why you do not get back to where you started. Finally, in the margin, write forty percent in all three notations.
Your rise-then-fall result should be below the original. If it came back to the start, check whether you applied the second percent to the new value or to the old one.
Recap
Five things, and the third one is where nearly every real percent problem is won or lost.
| If the question says | Your first move is |
|---|---|
| What is 30% of 70 | Convert 30% to 0.30, then multiply |
| 14 is 25% of what | Divide 14 by 0.25 |
| 8 is what percent of 20 | Divide 8 by 20, then multiply by 100 |
| Find the discount percent | Use the original price as the base |
| A 20% rise then a 20% fall | Multiply by 1.20 then by 0.80 |
That completes Chapter 3. Chapter 4 turns from solving equations to drawing them: plotting points on a coordinate plane and graphing the linear equations you have spent this chapter solving.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.9 Percents §3.9, pp. 183-188 — everything on these slides traces back here
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