Lesson 57: Rate Word Problems

New Concept

A rate shows a relationship between two different measurements.

What’s next

Next, you’ll practice solving rate problems using tables and multiplication formulas to find unknown quantities like distance or earnings.

Property

A rate shows a relationship between two different measurements. The phrase 'per hour' means 'in each hour'. We can make a table that shows how many miles the car travels in 1, 2, 3, and 4 hours. Word problems about rates have the same plot as 'equal groups' problems.

Example

• A baker decorates 10 cookies per minute. In 5 minutes, they decorate $10 \times 5 = 50$ cookies.
• A garden snail moves 2 centimeters per minute. After 8 minutes, it has traveled $2 \times 8 = 16$ centimeters.

Explanation

Think of a rate as a secret rule for how things change together! If a car's rate is 30 miles per hour, it means for every hour on the clock, the car travels exactly 30 miles. It's a super predictable pattern you can use to find future amounts, just like leveling up in a game.

Property

To solve a rate problem, use the formula: Number in each time group $\times$ Number of time groups = Total. This helps calculate the total amount when you know the rate and the duration. You can also switch the first two parts: Number of time groups $\times$ Number in each time group = Total.

Example

• Earning 12 dollars per hour for 4 hours of chores: $12 \text{ dollars per hour} \times 4 \text{ hours} = 48 \text{ dollars}$.
• A machine prints 50 t-shirts per hour. In an 8-hour shift, it prints $50 \times 8 = 400$ t-shirts.

Explanation

This is your ultimate shortcut for rate problems! Just plug in the numbers: find the amount for one group (like miles per hour), multiply by how many groups you have (the hours), and voilà, you get the grand total. It turns tricky word problems into simple math, making you a problem-solving wizard.

Property

A plan that can help us solve rate word problems is to make a table. We do this by writing the numbers we know into a table with two columns, like 'Weeks' and 'Dollars'. Then we can find the pattern and extend it. This method helps visualize the relationship and check your thinking.

Example

• Problem: A plant grows 3 inches per week. How much in 5 weeks? The table shows Week 1 -> 3 inches, Week 2 -> 6 inches. The pattern is $\times 3$. So, $5 \text{ weeks} \times 3 \text{ inches per week} = 15 \text{ inches}$.
• Problem: A cat naps 4 hours a day. How many hours in a week (7 days)? Table: Day 1 -> 4 hrs, Day 2 -> 8 hrs. The pattern is $\times 4$. So, $7 \text{ days} \times 4 \text{ hours per day} = 28 \text{ hours}$.

Explanation

Feeling stuck? A table is your visual superpower! By writing down the rate for 1, 2, and 3 groups, you can clearly see the multiplication pattern. It’s a great way to understand the problem before using the formula, making sure you’re on the right track to finding the correct answer.