Solve Proportions Worksheet Easily

Understanding Proportions and How to Solve Them

Proportions are a fundamental concept in mathematics, used to describe the relationship between two quantities. A proportion is a statement that two ratios are equal. It is often used to solve problems involving equivalent ratios, scaling, and proportionality. In this article, we will explore how to solve proportions easily, providing you with the necessary tools and techniques to tackle problems with confidence.

What is a Proportion?

A proportion is a statement that two ratios are equal. It is often represented using the symbol “∝” or “::”. For example, if we have two ratios, a:b and c:d, we can write them as a proportion:

a:b ∝ c:d

This means that the ratio of a to b is equal to the ratio of c to d.

How to Solve Proportions

Solving proportions involves finding the missing value in a proportion. Here are the steps to follow:

1. Write the proportion: Start by writing the proportion with the given values.
2. Identify the missing value: Determine which value is missing and needs to be solved for.
3. Use cross-multiplication: Multiply the numerator of the first ratio by the denominator of the second ratio, and vice versa. This will give you two equations.
4. Solve for the missing value: Solve for the missing value by dividing or multiplying the two equations.

Example 1: Simple Proportion

Solve the proportion:

2:3 ∝ x:6

To solve this proportion, we need to find the value of x.

📝 Note: To solve this proportion, we can use cross-multiplication. Multiply the numerator of the first ratio (2) by the denominator of the second ratio (6), and vice versa.

2 × 6 = 12 3 × x = 12

Now, divide both sides by 3:

x = 12 ÷ 3 x = 4

Therefore, the value of x is 4.

Example 2: Proportion with Multiple Variables

Solve the proportion:

x:5 ∝ 3:y

To solve this proportion, we need to find the values of x and y.

📝 Note: To solve this proportion, we can use cross-multiplication. Multiply the numerator of the first ratio (x) by the denominator of the second ratio (y), and vice versa.

x × y = 5 × 3 x × y = 15

Now, we can divide both sides by 5:

x = 15 ÷ 5 x = 3

Now that we have the value of x, we can substitute it back into the original proportion to solve for y:

3:5 ∝ 3:y

Using cross-multiplication again, we get:

3 × y = 5 × 3 3 × y = 15

Now, divide both sides by 3:

y = 15 ÷ 3 y = 5

Therefore, the values of x and y are 3 and 5, respectively.

Example 3: Real-World Application

A bakery sells a total of 250 loaves of bread per day. They offer a discount of 10% for bulk purchases. If a customer wants to buy x loaves of bread, and the bakery offers a discount of $0.50 per loaf for bulk purchases, how many loaves of bread must the customer buy to get the discount?

Let’s say the ratio of loaves of bread to the discount is x:0.50. We know that the bakery sells 250 loaves of bread per day, so we can set up the proportion:

x:0.50 ∝ 250:y

To solve this proportion, we need to find the value of x.

📝 Note: To solve this proportion, we can use cross-multiplication. Multiply the numerator of the first ratio (x) by the denominator of the second ratio (y), and vice versa.

x × y = 250 × 0.50 x × y = 125

Now, we can divide both sides by 0.50:

x = 125 ÷ 0.50 x = 250

Therefore, the customer must buy 250 loaves of bread to get the discount.

Conclusion

Solving proportions can be a straightforward process if you follow the steps outlined above. By using cross-multiplication and solving for the missing value, you can easily solve proportions and apply them to real-world problems. Remember to identify the missing value, use cross-multiplication, and solve for the missing value. With practice, you’ll become proficient in solving proportions and be able to tackle more complex problems with confidence.

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