Lesson Objectives

- Demonstrate an understanding of ratios and rates
- Demonstrate an understanding of the equality test for fractions
- Demonstrate an understanding of proportions
- Learn how to solve proportion equations
- Learn how to solve a proportion word problem

## How to Solve Proportion Equations

### Ratios

In our pre-algebra course, we learned about ratios and rates. Recall that a ratio is a comparison of two quantities. As an example, suppose a sack of marbles contains 30 red marbles and 20 green marbles. We can say the ratio of red marbles to green marbles is » 30:20. We can also write the ratio as a fraction: $$\frac{30}{20}$$ We can always simplify a ratio by dividing each part by the GCF. Here we can divide each part by the GCF (10). This will give us a simplified ratio of » 3:2. Again, this can also be shown as a fraction: $$\frac{3}{2}$$ This means for every 3 red marbles we have 2 green marbles. Again a ratio is nothing more than a comparison. In our scenario, we are just comparing the number of red marbles present to the number of green marbles present.### Unit Rate

Recall that a rate is a ratio where the units are different. When we think about rates, we most often come across the unit rate. The is the amount of some quantity per 1 unit of another. A typical example would be the number of miles a car can drive per one gallon of gas. As an example, suppose a new hybrid car can travel 819 miles on 13 gallons of gas. We can set up our rate as a fraction: $$\frac{819 \hspace{.25em}miles}{13 \hspace{.25em} gallons}$$ If we divide the number in the numerator (819) by the number in the denominator (13), we will get the unit rate: $$\require{cancel} \frac{63 \cancel{819} \hspace{.25em}miles}{\cancel{13} \hspace{.25em} gallons} = \frac{63 \hspace{.25em}}{1 \hspace{.25em} gallon}$$ This tells us our hybrid gets 63 miles per one gallon of gas.### Proportions

A proportion states that two ratios or two rates are equal. This can be useful for scaling up or down. Let’s suppose we had a recipe that calls for 3 eggs and 1 cup of sugar. If we wanted to double the recipe, we could simply double each part. $$\frac{3}{1} \cdot \frac{2}{2} = \frac{6}{2}$$ This would give us 6 eggs and 2 cups of sugar. These two ratios 3:1 and 6:2 represent a proportion.If we want to check to see if two ratios represent a proportion, we use the equality test for fractions. The equality test for fractions tells us that two fractions are equal if their cross products are equal. Cross products are formed by multiplying the denominator of one fraction by the numerator of the other. As an example, if we form the cross products of 3/1 and 6/2, we would see they are equal: 1 x 6 = 6

2 x 3 = 6

### Solving Proportion Equations

When we encounter proportion equations, we can solve the equation by cross multiplying first, and then performing our normal steps to solve an equation. Let's take a look at a few examples.Example 1: Solve each equation $$\frac{2}{5}=\frac{x}{10}$$ Cross Multiply: $$5x = 20$$ Solve the Equation: $$\frac{\cancel{5}}{\cancel{5}}x = \frac{4\cancel{20}}{\cancel{5}}$$ $$x = 4$$ Check: $$\frac{2}{5} = \frac{4}{10}$$ $$\frac{2}{5} = \frac{2\cancel{4}}{5\cancel{10}}$$ $$\frac{2}{5} = \frac{2}{5}$$ Example 2: Solve each equation $$\frac{1}{2} = \frac{2x - 5}{3}$$ Cross Multiply: $$2(2x - 5) = 3$$ Solve the Equation: $$4x - 10 = 3$$ $$4x = 13$$ $$\frac{\cancel{4}}{\cancel{4}}x = \frac{13}{4}$$ $$x = \frac{13}{4}$$ Check: $$\frac{1}{2} = \frac{2 \cdot \frac{13}{4} - 5}{3}$$ $$\frac{1}{2} = \frac{\cancel{2} \cdot \frac{13}{2 \cancel{4}} - 5}{3}$$ $$\frac{1}{2} = \frac{\frac{13}{2} - \frac{10}{2}}{3}$$ $$\frac{1}{2} = \frac{\frac{3}{2}}{3}$$ $$\frac{1}{2} = \frac{\cancel{3}}{2} \cdot \frac{1}{\cancel{3}}$$ $$\frac{1}{2} = \frac{1}{2}$$ Example 3: Solve each equation $$\frac{5x + 1}{6} = \frac{3x - 2}{3}$$ Cross Multiply: $$3(5x + 1) = 6 (3x - 2)$$ Solve the Equation: $$15x + 3 = 18x - 12$$ $$-3x = -15$$ $$\frac{\cancel{-3}}{\cancel{-3}}x = \frac{5 \cancel{-15}}{\cancel{-3}}$$ $$x = 5$$ Check: $$\frac{5(5) + 1}{6} = \frac{3(5) - 2}{3}$$ $$\frac{26}{6} = \frac{13}{3}$$ $$\frac{13 \cancel{26}}{3 \cancel{6}} = \frac{13}{3}$$ $$\frac{13}{3} = \frac{13}{3}$$

### Solving Proportion Word Problems

In many situations, we have a word problem that requires us to set up and solve a proportion equation. Let's take a look at an example.Example 4: Solve each word problem

The distance between Calymete Falls and Jacobsberg is 600 miles. On a given wall map, this distance is scaled down to 2.4 feet. On the same map, how many feet would be between Jacobsberg and Clydesville, two cities which are 1000 miles apart?

Let's set up our proportion. We know that 600 miles are scaled down to 2.4 feet on the map. We need to find out how many feet 1000 miles will be scaled down to on the same map. $$\frac{600 \hspace{.25em}miles}{2.4 \hspace{.25 em}feet} = \frac{1000 \hspace{.25em}miles}{x \hspace{.25em}feet}$$ We can drop the units and just work with the numbers. Let's cross multiply: $$2.4 \cdot 1000 = 600x$$ Solve the equation: $$2400 = 600x$$ $$\frac{4\cancel{2400}}{\cancel{600}} = \frac{\cancel{600}}{\cancel{600}}x$$ $$x = 4$$ This tells us the distance of 1000 miles between Clydesville and Jacobsberg would be represented on the map as 4 feet.

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