Unit Rate & Percent

What is a rate?

A rate compares two amounts that have different units, like dollars and pounds, or miles and hours. "3 pounds for 12 dollars" is a rate. It works like a ratio, but the two amounts are measured in different things.

A unit rate is the rate for one unit of the first quantity. If 3 pounds cost 12 dollars, then 1 pound costs 4 dollars. The 4 is the unit rate. You find it by dividing: 12 divided by 3 is 4.

Unit price and speed are both unit rates. "70 miles per hour" means 70 miles in 1 hour.

What is the double number line?

The module shows two number lines stacked on top of each other. The top line counts one quantity and the bottom line counts the other. Every vertical line hits a matching pair.

A handle slides along the lines. Move it and both lines move together, so the pair always keeps the same ratio. Set the ratio to 3:12 and the ticks read 3 pounds with 12 dollars, then 6 pounds with 24 dollars, and so on. The note at the bottom gives the unit rate: 1 pound = 4 dollars.

The table mode lists the same pairs as rows. It is the double number line written out as numbers.

What is a percent?

A percent is a rate per 100. So 30% means 30 out of every 100, or 30/100. In the module the top line runs from 0 to 100, so the top number is the percent. The bottom line is the whole quantity.

To find a percent of a quantity, line the handle up with the percent and read the bottom line. For example, 30% of 80 is 24. The readout says it in words: 30% of 80 = 24.

Percent is a rate per 100. 25% of 60 is 15, because 25/100 times 60 is 15.

Finding the whole from a percent

Sometimes you know the part and the percent, but not the whole. "30% of a number is 24" is that kind of problem.

One way is to find 1%. Divide the part by the percent: 24 divided by 30 is 0.8. So 1% is 0.8. Then multiply by 100 to get the whole: 0.8 times 100 is 80. The number is 80.

In the module, set the percent and then change the Quantity until the value on the right equals the part. When the value reads 24, the Quantity is the whole.

What they're learning

• How to read a unit rate from a double number line.
• How to use a table of equivalent ratios and fill in missing values.
• How to solve unit price and constant speed problems.
• How to find a percent of a quantity and how to find the whole from a percent.
• How to convert units by scaling the same ratio.

Aliases

• Unit rate, unit price
• Percent
• Double number line
• Ratio tables
• "Per" and "miles per hour"

Refresher

A rate compares two amounts with different units, like dollars and pounds. A unit rate is the amount for one unit of the first quantity. A percent is a rate per 100. A double number line lines up matching pairs so you can read the answer straight off.

Try this together

1. Open the module in Rate mode. Set the first term to 3 and the second term to 12. Ask what 1 pound costs and let them find the note at the bottom.
2. Switch to Percent. Set Quantity to 80 and Steps to 10. Drag to 30% and read the answer together.
3. Cover the readout and ask what 30% of 40 is before they drag the handle.
4. Switch to Convert. Set the ratio to 1:60 and ask how many minutes are in 4 hours.

If you get stuck, change one number at a time and watch the readout. You do not have to know the answer before you start.

Words you might hear

• Rate: a comparison of two amounts with different units, like miles per hour.
• Unit rate: the amount for one unit of the first quantity. 3 pounds for 12 dollars gives 4 dollars per pound.
• Percent: a rate per 100. 30% means 30/100.
• Whole: the full quantity a percent is taken from. In 30% of 80, the whole is 80.
• Equivalent ratios: rows that all keep the same ratio, like 3:5, 6:10, and 9:15.
• Scale: multiply both quantities by the same number so the ratio stays the same.

Deeper dive

If 3 pounds cost 12 dollars, then 1 pound costs 4 dollars, so the unit rate is 4 dollars per pound. The double number line shows 3 pounds across from 12 dollars, and 1 pound across from 4.

For percent, 30% of 80 is 24, because 30/100 × 80 = 24. If you know the part and the percent but not the whole, divide: 24 ÷ 30 is 0.8, so 1% is 0.8, and 0.8 × 100 is 80.