Lesson · Linear Equations and Inequalities · Applying equations

Ratio, Proportion and Percent

A ratio is a quotient, a proportion is an equation between two of them, and a percent is a ratio whose denominator is 100. Cross-multiplying is not a new rule: it is multiplying both sides by both nonzero denominators. The central work is keeping the two ratios in the same order and identifying the whole.

Open the interactive lesson Practice

a/b = c/d ⟺ ad = bc provided b ≠ 0 and d ≠ 0 x/4 = 3/8 → 8x = 12 → x = 3/2 part/whole = p/100 every percent question, once named

One identity, three uses

Core ideas

What this lesson establishes

These three topics are usually taught as three sets of rules. They are one equation, and the work is in setting it up rather than in solving it.

A ratio is a quotient and a proportion is an equation between two of them

a : b means the number a/b, which requires b ≠ 0. A proportion says two such numbers are equal, so it is an ordinary equation and every move you already have applies to it.

Cross-multiplying is one multiplication, not a rule of its own

Multiply both sides of a/b = c/d by bd and the denominators cancel, leaving ad = bc. Knowing that is what tells you when it is allowed: b and d must not be zero. Completion work here uses numerical denominators; variable-denominator equations belong to Rational and Radical Expressions.

Percent is a ratio with denominator 100

15% is the number 15/100, that is 3/20. Every percent question is part/whole = p/100 with one of the three quantities missing, and the only difficulty is deciding which number is the whole.

The material

Ratios, proportions and percents

What each one is, why cross-multiplication works, how units fix the ratio order, and how to identify the percent whole.

Definition · Ratio and proportion

A ratio of a to b, written a : b or a/b, is the quotient of the two, and requires b ≠ 0. A proportion is a statement that two ratios are equal.

A ratio compares two quantities of the same kind, or forms a rate from two of different kinds — kilometres per litre. The units decide what a correct set-up looks like.

Theorem · Cross products

If b ≠ 0 and d ≠ 0, then a/b = c/d is equivalent to ad = bc.

Since b ≠ 0 and d ≠ 0, the number bd is not zero. Multiplying both sides of a/b = c/d by bd gives (a/b)·bd = (c/d)·bd, which simplifies to ad = bc.

Dividing ad = bc by the nonzero number bd returns a/b = c/d, so the move is reversible and the two statements have the same solutions.

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The proof is where the conditions come from. In this lesson the denominators in the proportions you solve are stated nonzero numbers, so checking the condition is immediate. If a denominator contains the variable, multiplying by it may create a candidate at a value where the original quotient is undefined; that is a rational-equation issue, and Rational and Radical Expressions supplies the full method.

Setting one up

The two ratios must be built the same way round: if the left side is kilometres over litres, so is the right. A car covers 150 km on 12 L; how far on 18 L?

150/12 = d/18 km over litres on both sides × (12·18) 150·18 = 12d 2700 = 12d d = 225 check 225/18 = 25/2 and 150/12 = 25/2

Writing 150/12 = 18/d instead compares kilometres per litre with litres per kilometre. It is a different equation with a different answer, and the units are what tell you so before you solve it.

Percent

Definition · Percent

p percent, written p%, is the number p/100.

So 15% is 15/100 = 3/20, and "15% of 80" means (15/100)·80. The word is a unit of measurement for ratios, nothing more.

All three of the standard questions are the proportion part/whole = p/100 with a different letter missing:

Find the part. "What is 15% of 80?" — part/80 = 15/100, so 100·part = 1200 and the part is 12.
Find the percent. "42 is what percent of 120?" — 42/120 = p/100, so 120p = 4200 and p = 35.
Find the whole. "30 is 12% of what?" — 30/whole = 12/100, so 12·whole = 3000 and the whole is 250.

For a change rather than a share, the whole is the original amount: (new − old)/old = p/100. A price going from 60 to 75 is a rise of 15 on a base of 60, and 15/60 = 25/100, so 25%. Dividing by the new price instead gives 15/75 = 20/100 and answers a question nobody asked.

Example · Why percents do not cancel out

A price of 100 rises by 20% to 120, then falls by 20%. The fall is 20% of 120, which is 24, so the price lands at 96, not 100. The two percentages are taken on different bases. This is not a trick question — it is the reason a percent is meaningless until you say what it is a percent of.

Preview, not completion work. The proportion lab can also accept a variable in a denominator and can show a candidate that makes that denominator zero. Use those presets to preview Rational and Radical Expressions if you wish; they are not evidence that the domain method has been mastered here, and a quadratic cross product is outside this course.

Interactive

Proportions

Cross-multiplying, and what it assumes

Keep the ratio order, then cross-multiply

Before running the first or third preset, label what each numerator and denominator measures and predict the two cross products. Presets with a variable denominator are a preview of Rational and Radical Expressions, not completion work for this lesson.

Statement

Left numerator

Left denominator

Right numerator

Right denominator

cross product—

solutions—

excluded—

The two denominators may contain x. When they do, the values that make them zero are excluded before any solving happens.

Method

Solving a proportion

Most of the work is done before the algebra starts.

Say what each ratio compares, and put the two in the same order

Kilometres over litres on both sides, or part over whole on both sides. A proportion built out of order is a correctly solved answer to the wrong question.

Write down any value that would make a denominator zero

For this lesson both denominators are nonzero numbers, so this is a short check rather than a new solve. A variable denominator starts the rational-equation method deferred to Rational and Radical Expressions.

Multiply both sides by both denominators and solve

This is cross-multiplication, and it produces ad = bc. What follows is a linear equation of the kind “Variables on Both Sides” handles.

Check the candidate against the excluded list, then in the original

A candidate on the excluded list is not a solution and there may then be none. A candidate that survives still has to make the original proportion true, and substituting is what confirms it.

Proportions and percents

Score 0 / 0

Choose the answer you can justify.

A proportion, a rate, and the three percent questions

Every denominator here is a stated nonzero number; the hard step is keeping each comparison in the same order.

A. (x + 1)/3 = (x − 2)/5 denominators 3 and 5, never zero × 15 5(x + 1) = 3(x − 2) 5x + 5 = 3x − 6 2x = −11 x = −11/2 check left (−11/2 + 1)/3 = (−9/2)/3 = −3/2 right (−11/2 − 2)/5 = (−15/2)/5 = −3/2 agrees B. 8 notebooks cost 18; what do 14 cost? money/notebooks both times: 18/8 = c/14 cross products: 18·14 = 8c c = 63/2 check (63/2)/14 = 9/4 and 18/8 = 9/4 C. 15% of 80 part/80 = 15/100 100·part = 1200 part = 12 42 of 120 42/120 = p/100 120p = 4200 p = 35 30 is 12% of 30/w = 12/100 12w = 3000 w = 250

In B, 18/8 = 14/c would compare money per notebook on the left with notebooks per unit of money on the right. The units expose the reversal before any algebra does, which is why they belong beside the ratios.

In C the three lines are the same equation with a different unknown. Naming the whole is the entire difficulty: in the third line the whole is the thing being asked for, which is why 30 goes on top.

For a faded proportion, start 7/12 = n/30 from the supplied cross product 12n = 210. Finish the exact division and substitute into both ratios. Then say in words what would have to be measured by 7, 12, n and 30 for that set-up to be meaningful.

Common mistakes

Treating a variable-denominator preview as an ordinary proportionA denominator containing x can be zero for a candidate, so multiplying by it is not automatically reversible. The lab can display that warning, but Rational and Radical Expressions owns the exclusion-and-check method; completion here uses stated nonzero numerical denominators.

Building the two ratios in opposite orders150/12 = d/18 compares km per litre on both sides. 150/12 = 18/d compares km per litre with litres per km, and gives d = 36/25 — a clean answer to a question about the wrong quantity. Write the units beside each ratio.

Taking a percent of the wrong baseA 20% rise followed by a 20% fall leaves 96 out of 100, because the fall is 20% of 120. For a change, the base is always the original amount, and (new − old)/old says so explicitly.

Check your understanding

✓

Finish when you set the proportion up with matching units and identify the percent whole before you solve.

Without the lab, model this rate: 5 kg cost 14, so what do 8 kg cost? Label both ratio units, cross-multiply and check the exact answer. Then solve 45/w = 18/100 and say why w, rather than 45, is the whole. Variable-denominator equations remain the work of Rational and Radical Expressions.

Note: “Modelling with Linear Equations” takes the same step — turning a described situation into an equation — without the proportion to hang it on, which makes naming the unknown the main decision. The excluded-value habit built here returns in Rational and Radical Expressions, where every rational equation begins with it.