Chapter 4 · Ratios and rates
Compare Ratios with Tables and Spot Mistakes
Use ratio tables to compare two ratios by matching one quantity. Then check each row with one multiplier to catch mistakes made by adding instead of multiplying.
Picture two lemonade recipes. Recipe A uses 3 scoops of mix for every 4 cups of water. Recipe B uses 5 scoops of mix for every 8 cups of water. Which one tastes stronger?
You can't tell just by looking for the bigger numbers. Recipe B has more mix, but it also has more water.
To compare two ratios means to decide which one has more of one quantity for the same amount of the other quantity. You can compare these recipes because both count the same two quantities in the same units: scoops of mix and cups of water. So make one quantity match, then look at the other one. If neither has more, the ratios are equivalent.
A ratio table helps you find a match. You already know that every row of a correct ratio table shows an equivalent ratio. Here, each row is the same recipe in a different batch size. If both tables have a row with the same number of cups of water, you can compare the scoops of mix in those two rows fairly. If no rows match yet, write more rows of equivalent ratios until two rows do.
| Scoops of mix | Cups of water |
|---|---|
| Scoops of mix | Cups of water |
|---|---|
Worked example
Which lemonade tastes stronger?
Recipe A uses scoops of mix for every cups of water. Recipe B uses scoops of mix for every cups of water. Use the two tables above to decide which recipe tastes stronger.
Choose a quantity to match. Both tables have a row with 8 cups of water, so match the water. When the water is the same, only the scoops of mix can make one recipe taste stronger.
Read the row with 8 cups of water in Recipe A's table. It is the second row, which shows 2 batches, so it has 2 times each amount of the given ratio. The taste stays the same because 2 batches need 2 times the mix and 2 times the water.
Read the row with 8 cups of water in Recipe B's table. It is the first row, which is the given ratio, so it is correct.
The cups of water match, so compare the scoops of mix. Recipe A has 6 scoops and Recipe B has 5 scoops.
Say what the comparison means in this situation. More mix in the same 8 cups of water gives a stronger taste.
Answer
Recipe A tastes stronger. For the same cups of water, Recipe A uses scoops of mix and Recipe B uses only scoops.
Key concept
One multiplier for both quantities
In a ratio table, each column counts one quantity in one unit. Each row shows two amounts that go together. In a correct table, all the rows are equivalent ratios.
Let and be the two amounts in a correct row. Let be the multiplier: a number greater than zero that you multiply both amounts by. The new row is equivalent because equal groups hold times as much of each quantity.
- Compare: when two tables count the same two quantities in the same units, find correct rows where one quantity matches. Then compare the other quantity.
- Check: a row is correct when you can get it from a known correct row by multiplying both amounts by one multiplier. If the two columns need different multipliers, the row is a mistake.
A table only helps you compare if its rows are correct. A row has a mistake when it is not equivalent to the ratio that the table starts from.
First, say what each column counts and what one row shows. Then test each row against a row you know is correct. The ratio given in the problem is a correct row. In this lesson, every amount is greater than zero, so you can always divide.
- Pick one column. Divide the amount in the row you are testing by the amount in the correct row. Division undoes multiplication, so this gives the multiplier.
- Multiply the correct row's amount in the other column by that same multiplier.
- Compare your result with the table. If they are different, the row is a mistake.
Try it on Recipe A's second row, 6 scoops and 8 cups. The given ratio is 3 scoops to 4 cups. The multiplier for scoops is , because . Then cups, which matches the table. That row is correct.
| Red beads | Blue beads |
|---|---|
Worked example
Find the mistake in the bead table
Each bracelet uses red beads and blue beads. Mia chose the numbers of red beads first. Then she filled in the blue beads in the table above. Is every row correct? If not, which row is wrong, and what should it say?
Start from a row you know is correct. The first row is the given ratio: 3 red beads to 7 blue beads, which is 1 bracelet. Every other row must be this row with both amounts multiplied by one multiplier.
Test the second row. Divide its 6 red beads by 3 to find the multiplier, because division undoes multiplication. Then multiply the 7 blue beads by the same multiplier. The table says 14 blue beads, so this row is correct.
Test the third row. The multiplier for red beads is 3, so 9 red beads make 3 bracelets. Those 3 bracelets need 3 times 7 blue beads, which is 21. The table says 13, so this row is a mistake.
Test the fourth row too, so that you have checked every row. The multiplier is 5, and the table says 35 blue beads. This row is correct.
See how the mistake could have happened. The third row is the first row with 6 added to both amounts. But 6 more red beads make 2 more bracelets, and 2 more bracelets need 14 more blue beads, not 6 more.
Fix the row. Keep the 9 red beads that Mia chose. Replace 13 with 21, the number of blue beads that the multiplier 3 gives.
Answer
The third row is the mistake. With red beads there should be blue beads, not .
Now use both skills together. A comparison is only as good as the rows it uses. So check the rows you plan to compare before you compare them.
Theo wants to know which shop has the lower price for notebooks. Shop P sells notebooks in groups of 2 for 7 dollars. Shop Q sells them in groups of 3 for 10 dollars. Each shop charges the same amount for every group, no matter how many groups you buy. For each shop, Theo chose numbers of notebooks first and then filled in the costs.
| Notebooks | Cost in dollars |
|---|---|
| Notebooks | Cost in dollars |
|---|---|
Worked example
Check first, then compare
Shop P sells notebooks in groups of for dollars. Shop Q sells them in groups of for dollars. Use Theo's two tables above. Which shop has the lower price for the same number of notebooks?
Choose a quantity to match. Both tables have a row with 6 notebooks: the third row for Shop P and the second row for Shop Q. A wrong row would give a wrong comparison, so test both rows first.
Test Shop Q's row against its given ratio, 3 notebooks for 10 dollars. The multiplier for notebooks is 2, and the same multiplier gives the 20 dollars in the table. This row is correct.
Test Shop P's row against its given ratio, 2 notebooks for 7 dollars. The multiplier for notebooks is 3, so the cost must be 3 times 7 dollars. The table says 16 dollars, so this row is a mistake.
Name the mistake and fix it. The wrong row is the row above it, 4 notebooks for 14 dollars, with 2 added to both amounts. But 2 more notebooks are one more group, which costs 7 more dollars, not 2 more. Replace 16 with 21, the cost that the multiplier 3 gives.
Compare the two correct rows. The notebooks match at 6, so compare the dollars. Shop P charges 21 dollars and Shop Q charges 20 dollars.
Say what the comparison means in this situation. Fewer dollars for the same 6 notebooks is the lower price. The wrong row, with 16 dollars, would have made Shop P look cheaper, so the check changed the answer.
Answer
Shop Q has the lower price. For notebooks, Shop Q charges dollars and Shop P charges dollars.
Takeaway: in a correct ratio table, all the rows are equivalent ratios. You get one correct row from another by multiplying both amounts by one multiplier.
- To compare two ratios of the same two quantities in the same units, find correct rows where one quantity matches. Compare the other quantity, then say what that means in the situation.
- To find a mistake, test each row against the given ratio. If the two columns need different multipliers, the row is wrong. Look closely for rows where someone added the same number to both amounts.
Check first, then compare.