Ratios and Rates: Videos & Practice Problems

A ratio compares two quantities and can have the same or different units, or even no units at all. It is often written in words, as a fraction, or with a colon, and the order of the quantities matters. For example, the ratio of 9 women to 11 men can be written as 9:11 or $\frac{9}{11}$. A rate, on the other hand, is a specific type of ratio that compares two quantities with different units, such as miles per hour or dollars per week. Unlike ratios, the units in rates do not cancel out. For example, 420 miles in 8 hours is a rate and can be written as $\frac{\mathrm{420\; miles}}{\mathrm{8\; hours}}$. Understanding this distinction helps in solving problems involving comparisons of quantities with or without different units.

To simplify a ratio involving mixed numbers, first convert the mixed numbers into improper fractions. For example, if the ratio is 3 1/3 to 4 1/6, convert 3 1/3 to $\frac{10}{3}$ and 4 1/6 to $\frac{25}{6}$. Then, write the ratio as a fraction dividing the first by the second: $\frac{\frac{10}{3}}{\frac{25}{6}}$. Next, multiply by the reciprocal of the denominator fraction: $\frac{10}{3}\backslash (\backslash times\backslash )\frac{6}{25}$. Simplify common factors before multiplying. For instance, 6 and 3 share a factor of 3, and 10 and 25 share a factor of 5. After simplification, multiply the numerators and denominators to get the simplest form of the ratio. This process ensures the ratio is expressed with whole numbers in simplest form.

To find a unit rate from a given rate, divide the numerator by the denominator so that the denominator becomes 1. For example, if a person earns \(3600 in 6 weeks, the rate is $\frac{\mathrm{3600\; dollars}}{\mathrm{6\; weeks}}$. To find the unit rate (dollars per week), divide 3600 by 6: $3600\backslash div6=600$. So, the unit rate is \)600 per week, written as $\frac{\mathrm{600\; dollars}}{\mathrm{1\; week}}$. This method applies to any rate where you want to express the quantity per one unit of the second quantity, such as miles per hour or price per ounce.

The order of quantities in a ratio matters because it indicates which quantity is being compared to which. For example, the ratio of women to men in a class with 9 women and 11 men is written as 9:11 or $\frac{9}{11}$. If you reverse the order to men to women, it becomes 11:9 or $\frac{11}{9}$, which is a different ratio. This difference affects the interpretation and calculation of the ratio. Therefore, always pay attention to the order to accurately represent the relationship between the two quantities.

To calculate the unit price, which is a type of unit rate, divide the total price by the total quantity. For example, if a jar of salsa costs \(4.32 for 16 ounces, write the rate as $\frac{\mathrm{4.32\; dollars}}{\mathrm{16\; ounces}}$. Then divide 4.32 by 16 using long division or a calculator. The result is \)0.27 per ounce, written as $\frac{\mathrm{0.27\; dollars}}{\mathrm{1\; ounce}}$. This unit price helps compare costs across different quantities and is useful for making informed purchasing decisions.