Introduction to Rate Problems

A RATE is a type of ratio that compares two quantities with different units of mesure by division. A RATE expresses constant increase or decrease in one unit as compared to the increase or decrease in another, like distance over time or cost over weight. RATES can be useful tools to solve problems like the following:

Example: Christine scores 2 points on an excercise app for each 5 kilometers she runs. She wins a prize for every 100 points she gets. How many kilometers does she need to run in order to win a prize?

Christine's rate of points per kilometer is 2 points per 5 kilometers. So, the number of points she needs divided by the number of kilometers she must run to earn them will make an equivalent fraction.

Since Christine earns 2 points for every 5 kilometers and needs 2 points 50 times to get to 100, she also need to run 5 kilometers 50 times.

5 kilometers * 50 = 250 kilometers

x = 250 kilometers

Christine must run 250 kilometers to earn 100 points to and win a prize

Combining Rates

When two rates affect the same scenario at the same time, we combine the rates by adding or subtracting them. 

Example:  Lena uses 25 kilowatt-hours (kWh) of electricity each week, and her sister Marijke uses 31 kWh per week. How long will it be before the sisters uses 800 kWh in total?

Lena's rate of electricity uses is 25 kWh/week. Marijke's rate of electricity is 31 kWh/week. When we add the two rates together. we have combined rate:

Their combined rate of electricity usage is 56 kWh per week. This rate will be equal to the rate at which the sisters uses 800 kWh of electricity, so we can set the rates equal to each other and cross multiply to solve the unkonwn amount of time: 

It will take sisters 14 weeks and 2 days to use 800 kWh of electricity.

Now Lets try this on your own:

Jake can mow a lawn in 3 hours. Kelly can mow a lawn in 4 hours. If they work together, how long will it take them to mow one lawn?