Cross Multiplication

Accueil » Cross Multiplication
Cross Multiplication: meaning, method and solved example (7th Grade) – Maths PDF

In short: Cross multiplication turns an equation of two fractions into a plain equation by pairing opposite corners, so a/b = c/d becomes ad = bc.

7th GradeRatios and Percents7.RP.A.2c

What is Cross Multiplication?

Cross multiplication clears the fractions from an equation of two equal ratios. Multiply along each diagonal, set the two products equal, and the unknown stops hiding below a fraction bar.

Precisely, if \(\frac{a}{b} = \frac{c}{d}\) with \(b, d \neq 0\), then \(ad = bc\). It works because multiplying both sides by \(bd\) cancels the denominators. The trick needs exactly one fraction on each side.

How to spot it

  • An equation has one fraction on each side and a letter in one of the four positions.
  • A scaling, mixture or rate story has been written as a proportion.
  • You see a variable in a denominator and want to bring it up.
  • A check is needed that two ratios really match, for example two recipes scaled up by different cooks.

The method

  1. Write the equation with one fraction on each side.
  2. Multiply upper left by lower right, then lower left by upper right.
  3. Set the two products equal and solve the resulting equation.
  4. Substitute the answer into the original fractions and convert both to decimals or reduce them, so you can see that the two sides really match.
49=x454 x 45 = 9 x x gives 180 = 9x so x = 20
The two diagonal products 4 x 45 and 9 x x are equal.

Cross Multiplication: solved example

Problem

A lemonade stand in Tampa uses 6 lemons for every 15 cups of water. How many lemons are needed for 40 cups? Solve \(\frac{6}{15} = \frac{x}{40}\).

Solution

Multiply diagonally: \(6 \cdot 40 = 15x\), which is \(240 = 15x\).

Divide by 15: \(x = 16\). The stand needs 16 lemons. Check: \(\frac{6}{15} = 0.4\) and \(\frac{16}{40} = 0.4\).

Try it

Question 1

Solve \(\frac{5}{8} = \frac{35}{y}\).

Show the answer

\(5y = 280\), so \(y = 56\).

Question 2

Do \(\frac{3}{7}\) and \(\frac{12}{28}\) form a proportion? Test by cross multiplying.

Show the answer

\(3 \cdot 28 = 84\) and \(7 \cdot 12 = 84\). The products match, so yes.

Cross Multiplication vs. Proportion

A proportion is the equation made of two equal ratios. Cross multiplication is one method for solving it. You can also scale by a multiplier, using a table, without crossing at all.

Proportion

Keep going

Lessons

Worksheets and quizzes

Related words

Whole dictionary: A–Z · 7th Grade · Ratios and Percents

Voter.. page