# Solve Problems Involving Proportions: TEAS

## Basic Terms and Terminology Relating to Solving Problems Involving Proportions

- Ratio: The relationship of two numbers
- Proportion: Two ratios that are equal to each other

## Ratios and Their Meaning

Simply stated, a ratio is the relationship of two numbers and proportions are two ratios that are equal to each other.

The picture above is a ratio; this ratio could indicate that there are 4 boys for every 3 girls, that there are 4 pears for every 3 oranges or that there are $ 4 in the piggy bank for every 3 dollars in the drawer. As you can see with these examples, ratios give limited information. For example, a ratio does not tell you how many pears or oranges you actually have; a ratio does not tell you how many boys and how many girls there actually are; and, a ratio does not tell you how much money you have in the piggy bank or in the drawer.

Ratios are read as " 4 is to 3".

In order to determine how many pears or oranges you actually have, how many boys and how many girls there actually are and how much money you have in the piggy bank or in the drawer, you would have to perform ratio and proportion to answer these questions.

The different ways to express ratios are:

- 1/6
- 1:6
- 1 to 6

When comparing ratios, they should be written as fractions. The fractions must be equal. If they are not equal they are NOT considered a ratio. For example, the ratios 3/8 and 6/16 are equal and equivalent.

## Calculations Using Ratio and Proportion

Proportions are two ratios that are equal to each other and these ratio and proportion problems are calculated and solved as shown below.

**Example 1:**

If there is $12 in the drawer and the ratio of money in the drawer compared to money in the piggy bank is 4 :3, how much money is in the piggy bank?

4:3 = 12:x

**OR**

4/3 = 12x **OR** 12x = 4/3

12×3/4 = 36/4 = $9

Answer: There is $9 in the drawer.

**Example 2:**

If you have 8 oranges and the ratio of oranges compared to pears is 4 : 3, how many pears do you have?

4:3 = 8:x **OR** 4/3 = 8x **OR** 8x = 4/3

8×3/4 = 24/4 = 6

Answer: You have 6 pears.

**Example 3:**

If there are 24 boys in the group and there are 4 boys to 3 girls in the group, how many girls are in the group?

4:3 = 24:x **OR** 4/3 = 24x **OR** 24x = 4/3

24×3/4 = 72/4 = 18

Answer: There are 18 girls in this group.

Here are some ratio and proportion problems that entail different measurement systems and converting between different measurement systems:

**Example 1:**

Knowing that there are 2.2 pounds in one kilogram, how many kilograms will you weigh when your current weight is 156 pounds?

1.2 pounds : 1 kilogram = 156 pounds : x kilograms

2.2/1 = 156/x

2.2x/2.2 = 156/2.2

x = 156/2.2

x = 70.9 kg

Answer: You weigh 70.9 kilograms when you weigh 156 pounds.

**Example 2:**

Knowing that there are 2.2 pounds in one kilogram, how many pounds will you weigh when your current weight is 65 kilograms?

2.2 pounds : 1 kilogram = x pounds : 65 kilograms

2.2/1 = x/65

2.2x/2.2 = 65/2.2

x = 65 x 2.2

x = 143 pounds

Answer: You weigh 143 pounds when you weigh 65 kilograms.

**Example 3:**

Knowing that there are 60 drops in 1 teaspoon, how many teaspoons are in 74 drops?

60 drops : 1 teaspoon = 74 drops : x teaspoons

60/1 = 74/x

60x/60 = 74/60 = x

74/60 = 1.2 teaspoons

Answer: There are 1.2 teaspoons in 74 drops

Proportions are often used in the calculation of dosages and solutions in pharmacology and the preparation of medications by nurses, pharmacists and pharmacy technicians as well as us, in our everyday lives.

These different measurement systems will be fully discussed below in the section entitled "Measurement and Data: M 2; Objective 5 : Converting Within and Between Standard and Metric Systems", however for the moment, we would like you to see some of the most commonly used measurement system conversion factors.

**The most frequently used conversions are shown below. It is suggested that you memorize these.**

- 1 gr =60 mg
- 1 kg = 2.2 lb.
- 1 mg = 1,000 mcg
- 1 g = 1,000 mg
- 1 kg = 1,000 g
- 1 tbsp. = 3 tsp
- 1 tbsp. = 15 mL
- 1 tsp = 5 mL
- 1 l = 1,000 mL
- 1 oz. = 30 mL
- 1L = 1000 cc

## The Conversion of Percentages Into Ratios and Converting Ratios Into Percentages

The conversion of percentages into ratios can also be done.

The method for this is to place the percentage number and then : (colon) and then 100. A ratio is read as 12 is to 100 when you see 12 : 100, for example.

Here are some examples:

- 12% = 12 : 100 or 12 is to 100
- 120% = 120 : 100 or 120 is to 100
- 220% = 220 : 100 or 220 is to 100
- 2222% = 2222 : 100 or 2222 is to 100

Converting ratios into percentages is based on, again, the fact that ratios reflect parts of 100.

Here are some examples:

- 23 : 100 = 23% or 23 is to 100
- 567 : 100 = 567% or 567 is to 100
- 1,222 : 100 = 1,222% or 1,222 is to 100
- 32,678 : 100 = 32,678% or 32,678 is to 100
- 1 : 100 = 1% or 1 is to 100

## The Conversion of Fractions Into Ratios and Converting Ratios Into Fractions

As stated above, ratios can be expressed in three different ways as follows:

- 1/6
- 1:6
- 1 to 6

The conversion of fractions into ratios is done in the following manner. The numerator becomes the first number before the colon and the denominator is the number after the colon.

- 2/10

The ratio is 2 : 10 or 2 is to 10

- 23/56

The ratio is 23 : 56 or 23 is to 56

- 19/45

The ratio is 19 : 45 or 19 is to 45

- 2/99

The ratio is 2 : 99 or 2 is to 99

- 16/789

The ratio is 16 : 789 or 16 is to 789

- 1/1

The ratio is 1 : 1 or 1 is to 1

- 100/100

The ratio is 100 : 100 or 100 : 100

Here are some examples of converting percentages into word ratios:

- 123%

The ratio is 123 : 100

- 34%

The ratio is 34 : 100

- 1%

The ratio is 1 : 100

- 100%

The ratio is 100 : 100

- 1,222%

The ratio is 1,222 : 100

**RELATED TEAS NUMBERS & ALGEBRA CONTENT**:

- Converting Among Non Negative Fractions, Decimals, and Percentages
- Arithmetic Operations with Rational Numbers
- Comparing and Ordering Rational Numbers
- Solve Equations with One Variable
- Solve One or Multi-Step Problems with Rational Numbers
- Solve Problems Involving Percentages
- Applying Estimation Strategies and Rounding Rules for Real-World Problems
**Solve Problems Involving Proportions (Currently here)**- Solve Problems Involving Ratios and Rates of Change
- Translating Phrases and Sentences into Expressions, Equations and Inequalities