What is a Ratio?
A ratio is a way to show a comparison between two things, telling us how much of one thing there is compared to another. Imagine you’re baking cookies, and you need 2 cups of sugar for every 3 cups of flour. The ratio of sugar to flour would be written as 2:3. This means that for every 2 parts of sugar, you have 3 parts of flour. Ratios can also be written in fractions or in words.
- Colon:
- Fraction: $\frac{2}{
- Words: 2 to 3
Ratios are everywhere! For example, if you have 4 apples and 2 bananas, the ratio of apples to bananas is 4:2. You could also simplify it to 2:1 by dividing both numbers by 2. This tells us that for every 2 apples, there is 1 banana.
Ratios are often used in everyday situations, such as recipes, mixing colors, or comparing prices.
In the above figure, the ratio of squares to circles below is 3 to 4 or 3:4. This indicates that there are 3 squares to 4 circles. We can also say that the ratio of circles to squares is 4 to 3 or 4:3.
Antecedent and Consequent are the terms used to describe the two numbers that make up the ratio.
- Antecedent: The first number in a ratio.
- Consequent: The second number in a ratio.
Consider the ratio 4:5.
The number 4 is the antecedent because it comes first.
The number 5 is the consequent because it comes second.
Simplification of Ratios
Ratios can be simplified just like fractions, by cancelling out the common factors.
- If a classroom has 5 boys and 10 girls, the ratio of boys to girls is 5:10, which can be simplified to 1:2.
- If there are 4 blue marbles and 6 red marbles, the ratio of blue marbles to red marbles is 4:6, which can be simplified to 2:3.
Practice
Solve the following problems by identifying and simplifying the ratios.
- A fruit basket has 8 pears and 12 bananas. What is the ratio of pears to bananas?
- In a zoo, there are 25 zebras and 50 lions. What is the ratio of zebras to lions?
- A recipe calls for 3 cups of flour and 4 cups of sugar. What is the ratio of flour to sugar?
- A team has 5 boys and 15 girls. What is the ratio of boys to girls?
Equivalent Ratios
An equivalent ratio is like a recipe where the ingredients are scaled up or down but maintain the same proportions.
For example, imagine you have a recipe that uses 2 cups of flour and 3 cups of sugar. The ratio of flour to sugar is 2:3. If you wanted to make a bigger batch, you could double the ingredients, using 4 cups of flour and 6 cups of sugar. The ratio is 4:6, which is equivalent to 2:3 because when you simplify 4:6 by dividing both numbers by 2, you get 2:3.
Equivalent ratios are different sets of numbers that express the same relationship or proportion between two quantities. They can be found by multiplying or dividing both terms of the original ratio by the same non-zero number.
Ratios of Quantities with Different Units
When dealing with ratios of quantities that have different units but measure the same type of quantity, we need to convert one of the units so that both quantities are expressed in the same unit.
Let’s compare metres to centimetres. We know that 1 metre is equal to 100 centimetres.
If you have a ratio of 3 metres to 150 centimetres,
3 metres : 150 centimetres
Converting metres to centimetres, the ratio becomes
300 centimetres : 150 centimetres
On simplifying, 300:150 becomes 2:1. So, the ratio of 3 meters to 150 centimetres is equivalent to 2:1.
Finding a part of the total quantity
When you’re given a ratio and the total quantity, you can find a part of that quantity by using the ratio.
Example: In a garden, the ratio of red roses to yellow roses is 3:2. If there are 25 roses in total, how many red roses are there?
Step 1: Find the total number of parts.
- The ratio is 3:2. For every 3 red roses, we have 2 yellow roses.
- Total parts = 3 (red) + 2 (yellow) = 5 parts.
Step 2: Find the value of each part.
- Total number of roses = 25
- Value of each part = 25 ÷ 5 = 5 roses
Step 3: Calculate the specific part (number of red roses).
- Red roses = 3 parts × 5 roses = 15 red roses.
Now, what do you think is the number of yellow roses in the garden?
Practice
- A recipe calls for sugar and flour in the ratio of 4:5. If you have 180 grams of ingredients total, how much sugar do you need?
- The ratio of men to women in a company is 7:3. If there are 200 employees in total, how many women are there?
- A bag contains blue and green marbles in the ratio of 5:4. If there are 72 marbles in total, how many are green?
- In a park, the total number of swings and slides is 30. If the ratio of swings to slides is 3:2, find the number of slides in the park.
What is a Proportion?
A proportion is when two ratios are equal. It’s like saying two sets of things have the same relationship between them.
Example: Imagine you have two groups of pencils and pens:
- In the first group, you have 2 pencils for every 4 pens. So, the ratio is 2:4.
- In the second group, you have 3 pencils for every 6 pens. The ratio here is 3:6.
If you look at it, both groups have the same relationship between pencils and pens. This is because 2:4 is the same as 3:6. We say that these two ratios make a proportion.
Types of Proportion
Mean Proportion:
The mean proportion is like the “middle number” in a proportion. For example, in a proportion like , the mean proportion is the value of that makes both ratios equal. In this case, , so 4 is the mean proportion between 2 and 8.
Practice:
- Find the mean proportional between 9 and 25.
- If the mean proportional between two numbers is 12, and one of the numbers is 9, what is the other number?
- The mean proportion between two numbers is 15. If one of the numbers is doubled to become 36, what will be the new mean proportion?
Third Proportion:
If you have two numbers, like 6 and 12, the third proportion is the number that makes the proportion 6 : 12 = 12 : x true. Here, x = 24 because 6 : 12 = 12 : 24.
Practice:
- Find the third proportion to 8 and 16.
- If 5 and 10 are two numbers in proportion, find the third proportion.
- Find the third proportion to 7 and 14.
Fourth Proportion:
If you know three numbers, the fourth proportion is the number that makes the entire proportion work. For example, if you have 4 : 8 = 10 : x, the fourth proportion x = 20 because 4 : 8 is the same as 10 : 20.
Practice:
- If 3, 9, and 6 are in proportion, find the fourth proportion.
- For the numbers 7, 21, and 5, find the fourth proportion.
- A recipe uses the ratio of sugar to flour as 3:5. If you use 18 cups of sugar, how many cups of flour should be used to maintain the ratio?
Why Are Ratio and Proportion Important?
Understanding ratio and proportion is foundational for many other math concepts and real-life applications:
- Science and Technology: Proportions are used in scaling designs, mixtures, and even formulas.
- Art and Design: Ratios are essential in maintaining symmetry and creating visually pleasing compositions.
- Finance and Economics: Ratios help in comparing prices, interests, and other financial metrics.
- Everyday Life: Ratios and proportions are present in cooking, building, shopping, and more, making these concepts very practical.
FAQs on Ratio and Proportion
- A unit ratio is a ratio where the second term is 1, which helps compare quantities directly. For example, 3:1 is a unit ratio.
- An inverse ratio reverses the comparison of the original ratio. If the original ratio is a:b, the inverse ratio is b:a.
- In direct proportion, two quantities increase or decrease at the same rate. If one quantity doubles, the other doubles as well. The relationship can be written as a/b=c/d or a∝b.
- If 5 pencils cost $10, then 10 pencils will cost $20. Here, the number of pencils and the cost are in direct proportion because increasing the quantity of pencils increases the cost at the same rate.