We want to find out how we can know if two ratios are equivalent. Let's explore two situations that can help us understand the two most common ways.

Unit Rates

One dressing bottle has $6$ milliliters of oil for every $3$ milliliters of vinegar. Another bottle from the same brand has $18$ milliliters of oil for every $9$ milliliters of vinegar. We will write these ratios to check if they are the equivalent.

First Dressing Bottle	Second Dressing Bottle
$\frac{6 ml}{3 ml}$	$\frac{1 8 ml}{9 ml}$

We can use the fraction form of the ratios to find the units rate of each dressing bottle. Let's divide the numerator and denominator of the first ratio by

3 .

First Dressing Bottle \frac{6 \div 3 ml}{3 \div 3 ml} = \frac{2 ml}{1 ml}

We can follow a similar process to find the unit rate of the second dressing bottle. Let's divide the numerator and denominator of this ratio by

9 .

Second Dressing Bottle \frac{1 8 \div 9 ml}{9 \div 9 ml} = \frac{2 ml}{1 ml}

Since the ratios have the same unit rate, they are equivalent.

Equivalent Fractions

In Spanish La Liga, the best player scored $7$ penalties out of $13$ attempts. In contrast, the top player in the English Premier League scored $21$ out of $26$ attempts. We will compare the ratios of these players.

La Liga	English Premier League
$\frac{7}{1 3}$	$\frac{2 1}{2 6}$

When we multiply the numerator of the first ratio by

3

and its numerator by

2,

we obtain the second ratio.

\frac{7 \times 3}{1 3 \times 2} = \frac{2 1}{2 6}

However, we must multiply both the numerator and the denominator of a ratio by the same number to obtain an equivalent fraction. This means that in this case the ratios are not equivalent.

Conclusion

We can now state two ways of determining if two ratios are equivalent.

Finding the unit rate of the ratios.
Using equivalent fractions.

Note that there can be other ways to determine if two ratios are equivalent.

Exercise

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