Have you ever argued with a friend about who got the “better deal” on snacks or who has a higher scoring average in a game? Those arguments are all about comparisons, and comparisons are exactly what ratios are made for.
In this lesson, you will learn how to create tables of equivalent ratios, fill in missing values, and then plot those pairs on a coordinate plane. You will also learn how to use these tables to compare different ratios and decide which is bigger, faster, or better.A ratio compares two quantities. These quantities are often counts or measurements, like number of students, cups of water, miles, or dollars. In this lesson, we focus on ratios that use whole number measurements.
A ratio can be written in three main ways:
All three mean the same thing: we are comparing 3 of one thing to 2 of another thing.
Example: Suppose in a classroom there are 4 cats in pictures and 6 dogs in pictures. The ratio of cats to dogs is:
This ratio compares “number of cats” to “number of dogs.” The order matters. The ratio of dogs to cats is different: it is 6 to 4, or \(\dfrac{6}{4}\).
Ratios help answer questions like: “For every how many dogs, how many cats are there?” or “How many cups of juice are used with each cup of water?”
Two ratios are equivalent if they describe the same relationship, even if the numbers look different. This is similar to equivalent fractions. For example, \(\dfrac{1}{2}\) and \(\dfrac{2}{4}\) are equivalent fractions, because they represent the same amount.
The ratios 2:3 and 4:6 are equivalent ratios because both describe the same basic comparison.
We can create equivalent ratios by multiplying or dividing both parts of the ratio by the same non-zero number.
For a ratio \(a:b\), we can make an equivalent ratio by doing:
\[a:b \rightarrow ka:kb\]
where \(k\) is any positive number. For whole-number measurements, we usually use whole-number \(k\).
Example:
So 3:5 and 6:10 are equivalent ratios.
Another example: Divide both parts by 2.
The ratios 8:12 and 4:6 are equivalent.
A ratio table is a table that lists pairs of numbers that are all equivalent to the same ratio. It helps you see the pattern and quickly find missing values. As you will see in [Figure 1], each row of a ratio table shows a pair of numbers that keeps the same comparison.
Let’s build a ratio table step by step.
Suppose you are buying oranges. The store sells oranges at a constant rate: 2 oranges cost 1 dollar. The ratio of oranges to dollars is 2:1, or \(\dfrac{2}{1}\).
We can make a table with one column for oranges and one column for dollars.
Start with the basic ratio:
Now multiply both numbers by different whole numbers to get equivalent ratios:
If we arrange these in a table, we might have:
Oranges: \(2, 4, 6, 8, 10, ...\)
Dollars: \(1, 2, 3, 4, 5, ...\)
Notice that every time oranges go up by 2, dollars go up by 1. Every row is an equivalent ratio to 2:1.
You can build ratio tables for many situations: recipe ingredients, number of students and number of teams, distance and time, and more. The important idea is that each row is created by multiplying or dividing both columns by the same number.
Sometimes a ratio table is partly filled in, and you need to find the missing values. You can use multiplication or division, and sometimes both, to figure them out. This skill is powerful when dealing with real-world problems.
Method 1: Scaling from a known row
Suppose you know the ratio of pencils to boxes is 5:1. That means 5 pencils fit in 1 box.
Here is a partial ratio table:
Pencils: \(5, 10, 15, 25\)
Boxes: \(1, 2, 3, ?\)
We want to find the missing number of boxes when there are 25 pencils.
Step 1: Notice how to get from 5 pencils to 25 pencils.
\[5 \times 5 = 25\]
Step 2: Since we multiply pencils by 5, we must do the same to boxes:
\[1 \times 5 = 5\]
So 25 pencils need 5 boxes. The missing value is 5.
Method 2: Using division to go back to the basic ratio
Sometimes the table jumps to a large number, and dividing first is easier.
Example: The ratio of red beads to blue beads is 3:4.
Partial table:
Red beads: \(3, 6, 12, ?\)
Blue beads: \(4, 8, 16, 40\)
We want to find the missing number of red beads when there are 40 blue beads.
Step 1: Look at the blue beads. How do we get from 4 to 40?
\[4 \times 10 = 40\]
So the basic ratio was multiplied by 10.
Step 2: Multiply the red beads in the basic ratio by 10.
\[3 \times 10 = 30\]
So when there are 40 blue beads, there are 30 red beads. The missing value is 30.
Method 3: Using unit rate (per 1)
Sometimes it helps to figure out “for 1 of something, how many of the other?” This is called thinking about a unit rate.
Example: If 12 apples cost 6 dollars, then the ratio of apples to dollars is 12:6.
To find apples per 1 dollar, divide both numbers by 6:
\[\frac{12}{6} = 2, \quad \frac{6}{6} = 1\]
So the unit rate is 2 apples per 1 dollar (ratio 2:1).
Then if you want to know how many apples for 5 dollars, multiply both parts of 2:1 by 5:
\[2 \times 5 = 10, \quad 1 \times 5 = 5\]
So 5 dollars buys 10 apples.
A ratio table is not just a list of numbers. Each row in the table can also be seen as a point on a graph.
As you will see in [Figure 2], these points line up in a special way.The coordinate plane has two number lines:
An ordered pair \((x,y)\) tells us a point’s position: \(x\) units across, \(y\) units up.
To graph ratio pairs, we must decide which quantity goes on which axis. We usually put the first quantity in the ratio on the x-axis and the second quantity on the y-axis.
Example: Juice and water mix
Suppose the ratio of cups of juice to cups of water is 1:2. That means for every 1 cup of juice, we use 2 cups of water.
Make a ratio table:
Juice (cups): \(1, 2, 3, 4\)
Water (cups): \(2, 4, 6, 8\)
Now turn each row into an ordered pair \((x,y)\), where \(x\) is juice and \(y\) is water:
If we plot these points on a coordinate plane, they all lie on a straight line that goes through the origin \((0,0)\).
This happens for any table of equivalent ratios: when you plot the pairs, they line up on a straight line through the origin. This shows that the relationship is multiplicative, not just additive.
For example, we can say that water is always 2 times the amount of juice:
\[\textrm{water} = 2 \times \textrm{juice}\]
This is why the points are on a straight line. If you know one value (juice), you can multiply by 2 to find the other (water).
Imagine you have two different sports drink mixes, and you want to know which one has a stronger flavor. Comparing ratios fairly can be tricky, but tables make it much easier. As shown in [Figure 3], lining up ratios in tables helps you compare them for the same amount of one ingredient.
Example: Comparing drink strength
Drink A uses a ratio of 2 scoops of powder to 3 cups of water (2:3).
Drink B uses a ratio of 3 scoops of powder to 5 cups of water (3:5).
We want to know which drink has more powder in each cup of water, or which one tastes stronger.
One way is to make ratio tables and look for a common number of cups of water, then see which drink has more powder.
Drink A (2:3):
Drink B (3:5):
Now look for a way to compare. One simple method is to make the water the same. Let’s choose 15 cups of water.
For Drink A: Start from 3 cups of water. To get 15 cups:
\[3 \times 5 = 15\]
So multiply the powder by 5:
\[2 \times 5 = 10\]
So for Drink A, 15 cups of water need 10 scoops of powder (10:15).
For Drink B: We already have 9:15 in the table (9 scoops, 15 cups).
Now compare:
Drink A has more powder for the same water, so Drink A is stronger.
This method works for many situations:
The main idea is to use tables to line up ratios so that one of the quantities (like water, dollars, or time) is the same. Then you can compare the other quantity fairly.
Ratio tables and graphs show up all around you in real life. Here are some important ways they are used.
1. Cooking and recipes
If a recipe uses 3 cups of flour and 2 cups of sugar (ratio 3:2) to make a small batch of cookies, and you want to make a larger batch, a ratio table can help:
Flour (cups): \(3, 6, 9, 12\)
Sugar (cups): \(2, 4, 6, 8\)
If you choose 9 cups of flour, the table shows you need 6 cups of sugar to keep the flavor the same.
2. Speed and distance
If a car travels 60 miles in 1 hour, the ratio of distance to time is 60:1. A ratio table can tell you how far the car goes in other times:
Time (hours): \(1, 2, 3, 4\)
Distance (miles): \(60, 120, 180, 240\)
Each row is an equivalent ratio to 60:1. These points \((1,60), (2,120), (3,180), (4,240)\) would lie on a straight line through the origin on a distance-time graph.
3. Money and shopping
When you see “3 cans for 4 dollars,” that is a ratio of cans to dollars: 3:4.
A ratio table helps you find the cost of different numbers of cans:
Cans: \(3, 6, 9, 12\)
Dollars: \(4, 8, 12, 16\)
Using the table, you can see patterns and decide if a bigger pack is a better deal or not.
Example 1: Completing a ratio table
The ratio of girls to boys in a club is 4:5. Complete the table.
Girls: \(4, 8, 12, ?\)
Boys: \(5, 10, ?, 25\)
Step 1: Fill the third entry for boys.
Look at how we go from 4 girls to 12 girls:
\[4 \times 3 = 12\]
So we multiply by 3. Do the same to boys:
\[5 \times 3 = 15\]
So the missing number of boys in the third position is 15.
Now the table is:
Girls: \(4, 8, 12, ?\)
Boys: \(5, 10, 15, 25\)
Step 2: Find the missing number of girls when there are 25 boys.
Compare boys: from 5 to 25:
\[5 \times 5 = 25\]
So multiply girls by 5 too:
\[4 \times 5 = 20\]
The missing number of girls is 20.
Final table:
Girls: \(4, 8, 12, 20\)
Boys: \(5, 10, 15, 25\)
Example 2: Plotting points from a ratio table
A music app downloads 3 songs every 2 minutes. The ratio of songs to minutes is 3:2.
Step 1: Make a ratio table.
Minutes: \(2, 4, 6, 8\)
Songs: \(3, 6, 9, 12\)
Each row is an equivalent ratio to 3:2.
Step 2: Create ordered pairs \((x,y)\) with x = minutes, y = songs.
Step 3: Plot the points.
On the x-axis, label minutes. On the y-axis, label songs. When you plot the four points, they form a straight line through the origin. This shows that the number of songs is proportional to time.
Example 3: Using tables to compare ratios
Two runners are training:
Who is faster?
Step 1: Write each ratio as miles to minutes.
Step 2: Use ratio tables to find each runner’s distance in 8 minutes, or simplify to a smaller equivalent ratio.
For Runner A (5:40), divide both by 5:
\[\frac{5}{5} = 1, \quad \frac{40}{5} = 8\]
So Runner A’s ratio is equivalent to 1 mile in 8 minutes.
For Runner B (7:56), divide both by 7:
\[\frac{7}{7} = 1, \quad \frac{56}{7} = 8\]
So Runner B’s ratio is also equivalent to 1 mile in 8 minutes.
Both runners have the same equivalent ratio: 1 mile per 8 minutes. Using the idea of ratio tables or simplifying, we see that they are running at the same speed.
🎯 A ratio compares two quantities. It can be written as “a to b,” “a:b,” or \(\dfrac{a}{b}\).
🎯 Equivalent ratios are made by multiplying or dividing both parts of a ratio by the same non-zero number. They describe the same relationship.
🎯 A ratio table lists pairs of numbers that are all equivalent ratios. Each row in the table keeps the same comparison between the two quantities.
🎯 To find missing values in a ratio table, use multiplication and division. You can scale from a known row or find the unit rate (per 1) and then scale up.
🎯 Ratio pairs can be graphed on a coordinate plane by turning each row into an ordered pair \((x,y)\). For a proportional relationship, the points lie on a straight line through the origin.
🎯 Tables of equivalent ratios are powerful tools for comparing different ratios fairly by lining them up with a common value for one of the quantities.
🎯 These ideas show up in many real-world situations: recipes, shopping, travel speed, sports statistics, and more. Understanding ratio tables and graphs helps you make smart decisions in everyday life.
