Proportions

A proportion is an equation stating that two ratios or rates are equal. It is written in the following form.

Proportions are mathematical relationships between two equal ratios. In other words, proportions express the equality of two ratios. They are used to compare quantities and find unknown values. Proportions are everywhere in our daily lives, from cooking recipes to financial calculations to geometry.

Setting Up Proportions

   - Proportions involve setting two ratios equal to each other.

   - If \( \frac{a}{b} = \frac{c}{d} \), where \( b \) and \( d \) are not zero, then \( a \) and \( c \) are in proportion.

   - For example, if it takes 2 hours to travel 60 miles, then to find out how long it will take to travel 120 miles,

we can set up the proportion: \( \frac{2}{60} = \frac{x}{120} \).

Solving Proportions

   - To solve a proportion, cross multiply and then solve for the unknown variable.

   - For example, in the proportion \( \frac{2}{60} = \frac{x}{120} \),

cross multiplying gives \( 2 \times 120 = 60 \times x \). Solving for \( x \) gives \( x = \frac{2 \times 120}{60} = 4 \) hours.

Proportional Relationship

   - Proportional relationships are relationships between two quantities where the ratio of one quantity to the other remains constant.

   - For example, if a car travels at a constant speed, the distance it travels is directly proportional to the time it travels.

Examples and Practice

1. If 4 lemons cost $2, how much do 7 lemons cost?

   Let the cost of 7 lemons be \( x \) dollars.

   We can set up the proportion: \( \frac{4}{2} = \frac{7}{x} \).

   Cross multiplying gives: \( 4x = 2 \times 7 \).

   Solving for \( x \): \( x = \frac{2 \times 7}{4} = 3.5 \) dollars.

   So, 7 lemons cost $3.50.

2. A map has a scale of 1 inch represents 20 miles. If two cities are 3.5 inches apart on the map, how far apart are they in reality?

   Let the actual distance between the cities be \( d \) miles.

   We can set up the proportion: \( \frac{1}{20} = \frac{3.5}{d} \).

   Cross multiplying gives: \( 1 \times d = 20 \times 3.5 \).

   Solving for \( d \): \( d = \frac{20 \times 3.5}{1} = 70 \) miles.

   So, the cities are 70 miles apart in reality.

3. A recipe calls for 3 cups of flour to make 24 cookies. How many cups of flour are needed to make 36 cookies?

   Let \( x \) be the number of cups of flour needed for 36 cookies.

   We can set up the proportion: \( \frac{3}{24} = \frac{x}{36} \).

   Cross multiplying gives: \( 3 \times 36 = 24 \times x \).

   Solving for \( x \): \( x = \frac{3 \times 36}{24} = 4.5 \) cups.

   So, 4.5 cups of flour are needed to make 36 cookies.

4. A car travels 240 miles in 4 hours. How long will it take to travel 360 miles at the same speed?

   Let \( t \) be the time it takes to travel 360 miles.

   We can set up the proportion: \( \frac{240}{4} = \frac{360}{t} \).

   Cross multiplying gives: \( 240 \times t = 4 \times 360 \).

   Solving for \( t \): \( t = \frac{4 \times 360}{240} = 6 \) hours.

   So, it will take 6 hours to travel 360 miles at the same speed.