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Proportions (Solving)

Proportions are two fractions (that sometimes represent ratios) that are equal to each other.  Proportions are very helpful because you know that they are equal fractions.  If you know that the fractions are equal you can find missing pieces of information by cross-multiplying.  In the next lesson, we'll talk more about creating proportions, for now, let's look at how to solve them. 

When you have a proportion, you can cross multiply to create an algebraic equation that you can solve.

You can identify a proportion easily: it's two fractions joined by an equal sign.

If you have $\dfrac{6}{7}=\dfrac{12}{14}$, you have a proportion: two fractions that are equal to each other.

This comes in handy, when you're missing either the numerator or the denominator of one of the fractions.


$\dfrac{4}{5}=\dfrac{x}{35}\text{, what is the value of }x\text{?}$

Whenever you have a proprtion you can cross multiply to create an algebraic equation you can use to solve for the variable.  Cross multiply means to multiply the numerator of one fraction by the denominator of the other fraction, and then set that term equal to the term you find by multiplying the other numerator times the other denominator.  Let's try it. 

$$\eqalign{\dfrac{4}{5}&=\dfrac{x}{35}\\\dfrac{4}{5}&\searrow\dfrac{x}{35}\qquad&&\text{Multiply: }4\times35=140\\\dfrac{4}{5}&\swarrow\dfrac{x}{35}\qquad&&\text{Multiply: }x \times 5=5x\\140&=5x\qquad&&\text{Set the two multiplication answers equal to each other}\\\div 5 &=\div 5 \qquad&&\text{Solve for x by dividing both sides by 5}\\28&=x\qquad&&\text{You found that x=28}}$$


Any time you have a proportion, or two equal fractions, you can cross multiply to solve for a variable.  You can also use the fact that cross-products in proportions are equal if the two fractions are equal to prove that two fractions are equal!


Are the two fractions $\dfrac{6}{21}$ and $\dfrac{.4}{1.4}$ equal?

Set them up as a proportion, cross-multiply, and see.  If they are equal, their cross products will be equal.

$$\eqalign{\dfrac{6}{21}&=\dfrac{.4}{1.4}\\\dfrac{6}{21}&\searrow\dfrac{.4}{1.4}\qquad&&6 \times 1.4=8.4\\\dfrac{6}{21}&\swarrow\dfrac{.4}{1.4}\qquad&&.4\times21=8.4\\8.4&=8.4\qquad&&\text{The cross products are equal so the fractions are equal}}$$

Overall, when you have two fractions that are equal, their cross-products are equal. You can use that rule to solve for missing variables or to check to see if fractions are equal. 

Practice Problems:

  • Proportions (Solving)

    Solve the proportion.

    1. $\dfrac{n}{12}=\dfrac{3}{4}$

    2. $\dfrac{50}{20}=\dfrac{z}{16}$

    3. $\dfrac{25}{3}=\dfrac{t}{51}$

    4. $\dfrac{6}{c}=\dfrac{54}{99}$

    5. $\dfrac{n}{14}=\dfrac{63}{84}$

    6. $\dfrac{2.1}{0.9}=\dfrac{27.3}{y}$

    7. $\dfrac{16.2}{67.4}=\dfrac{x}{134.8}$

    8. $\dfrac{8}{a}=\dfrac{0.4}{0.62}$

    9. $\dfrac{b}{1.8}=\dfrac{49.6}{14.4}$

    10. $\dfrac{2}{3}=\dfrac{4}{z}$

    11. $\dfrac{6}{a}=\dfrac{3}{1}$

    12. $\dfrac{39}{13}=\dfrac{9}{d}$

    Answer Key:

Test Prep Practice

  • Pre Algebra: Proportions

    1. If $\dfrac{x}{16}=\dfrac{12}{y}$, then what is the value of $xy$?

    (A) $\dfrac{3}{4}$
    (B) $144$
    (C) $192$
    (D) $\dfrac{4}{3}$
    (E) $224$

    2. If $\dfrac{a}{8}=\dfrac{1}{16}$, then what is the value of $a$?

    (A) $\dfrac{1}{2}$
    (B) $128$
    (C) $8$
    (D) $16$
    (E) $2$


    3. If $\dfrac{12}{n-2}=\dfrac{8}{2n}$, then what is the value of $n$?

    (A) $-1$
    (B) $1$
    (C) $\dfrac{1}{2}$
    (D) $\dfrac{-1}{2}$
    (E) $-2$

    4. The ratio of 1.8 to 3 is not equal to which of the following ratios?

    (A) 18 to 30
    (B) 3 to 10
    (C) 6 to 10
    (D) 9 to 15
    (E) 0.6 to 1