Chapter 4 · Ratios and rates
Scale Ratios and Plot Their Pairs
Solve word problems by scaling both quantities in a ratio by the same factor. Then plot equivalent ratios as points and see them line up on a coordinate plane.
A ratio compares two quantities. It tells you how much of one goes with how much of the other. Say a green paint recipe uses cups of blue paint for every cups of yellow paint. The ratio of blue to yellow is .
Now make a double batch: cups of blue and cups of yellow. The bucket holds two full copies of the recipe, so every cups of blue still go with cups of yellow. The green is the same shade. Ratios that describe the same relationship, such as and , are equivalent ratios.
In this lesson, you will use equivalent ratios to solve word problems. Then you will turn them into points on a grid and see what those points have in common.
Key concept
One scale factor for both quantities
Equivalent ratios show the same relationship at different sizes. To make one, multiply both quantities by the same number greater than . That number is the scale factor. In the formula, and are the two quantities in a ratio and is the scale factor, with .
Why does the relationship stay the same? Multiplying both quantities by makes full copies of the original pair. Each copy holds the same amounts, so the relationship is the same at every size.
Dividing both quantities by the same number greater than works too. Dividing by undoes multiplying by and gives back one copy.
Worked example
Scale a recipe up
A green paint recipe uses cups of blue paint for every cups of yellow paint. You pour in cups of blue paint. How many cups of yellow paint do you need to make the same green?
Write the recipe as a ratio and fix the order: blue first, yellow second. Keeping one order stops you from mixing up the two quantities.
Find the scale factor. Blue goes from 2 cups to 8 cups. Divide 8 by 2 to find the number that turns the recipe amount into the new amount.
Multiply yellow by the same scale factor: 3 times 4 is 12. Using 4 for both quantities makes 4 full copies of the recipe, so the shade does not change.
Check by dividing both new amounts by 4. Division undoes the multiplication, so you should get the recipe back.
Answer
You need cups of yellow paint, because is equivalent to .
Worked example
Scale down first, then up
A lemonade recipe uses cups of water for every lemons. You have lemons. How many cups of water do you need to keep the same taste?
Write the ratio with water first and lemons second, so each number keeps its meaning.
No whole number times 4 gives 10, so first scale the recipe down to smaller numbers. Divide both quantities by 2. This splits the recipe into 2 equal half-batches, and each half-batch keeps the same taste: 3 cups of water for every 2 lemons.
Find the scale factor from 2 lemons to 10 lemons. Divide 10 by 2, because the scale factor is the number that multiplies 2 to make 10.
Multiply the water by the same scale factor, because both quantities must grow together: 3 times 5 is 15.
Check against the original recipe. Dividing by 2 and then multiplying by 5 is the same as multiplying by 2.5. To multiply by 2.5, double the amount and add half of it. Both original amounts times 2.5 give the new amounts, so one scale factor links the two ratios.
Answer
You need cups of water for lemons, because is equivalent to .
You can also turn equivalent ratios into a picture. A coordinate plane is a grid with two number lines: the horizontal -axis and the vertical -axis. They cross at the origin, the point . An ordered pair names one point on the grid. In this lesson every amount is or more, so you reach a point by starting at the origin, moving right units, and then moving up units.
To plot ratios, give each quantity its own axis and keep that choice for every pair. For the green paint, let be the number of cups of blue paint and let be the number of cups of yellow paint.
The ratio table below lists equivalent ratios, one in each row, like the tables you have completed before. Each row is one batch of the same green. The first column counts cups of blue paint. The second column counts cups of yellow paint. The third column writes the row as an ordered pair.
| Cups of blue paint (x) | Cups of yellow paint (y) | Ordered pair (x, y) |
|---|---|---|
Worked example
Plot the paint ratios
Plot the four batches from the ratio table on a coordinate plane. Use the -axis for cups of blue paint and the -axis for cups of yellow paint. What do the four points have in common?
Set up the axes so that every pair fits. The largest amounts are 8 cups of blue and 12 cups of yellow. An x-axis from 0 to 10 and a y-axis from 0 to 15, with grid lines 1 cup apart, has room for all four pairs.
Plot the first row. Start at the origin, move right 2 for the 2 cups of blue, then move up 3 for the 3 cups of yellow.
Plot the other three rows the same way. Always move right for blue first and then up for yellow, because the order in an ordered pair matters.
Compare each point with the one before it. Each row holds one more copy of the recipe than the row above it: 2 more cups of blue and 3 more cups of yellow. So every step goes right 2 and up 3, and equal steps stay on one straight line.
Take one step back from the first point: 2 left and 3 down. You land on the origin, so the straight line through the four points passes through the origin too.
Answer
The points , , and lie on one straight line, and that line also passes through the origin, .
In the figure above, the -axis counts cups of blue paint and the -axis counts cups of yellow paint. The grid lines are cup apart. Every amount is a whole number of cups, so each point sits exactly where two grid lines cross. For example, the point is the batch with cups of blue and cups of yellow. Line A passes through all four points and through the origin. The origin means cups of blue with cups of yellow. An empty bucket is not a batch, so is on the line but is not one of the ratios.
Plot any set of equivalent ratios this way, with the same quantity on the -axis for every pair, and the points lie on one straight line through the origin. Scaling both quantities by one scale factor slides a point along that line, never off it. So if a pair's point is not on that line, the pair is not equivalent to the others.
Here is the idea to keep. Equivalent ratios are the same relationship at different sizes. In a word problem, find the scale factor for one quantity, then multiply the other quantity by the same scale factor. When you plot equivalent ratios, with the same quantity on the -axis every time, the points lie on one straight line through . Next, you will use ratio tables to compare ratios and to spot mistakes in tables.