Study Guides | ratios and proportions | Ratios

Study Guide: Ratios

A ratio compares two or more values.

A ratio compares two or more values. A ration shows how much of one thing there is compared to another thing.

For example, if there are eight oranges (🍊🍊🍊🍊🍊🍊🍊🍊) and six lemons (🍋🍋🍋🍋🍋🍋) in a bowl of fruit, then the ratio of oranges to lemons is eight to six ($8 ∶ 6$), which is also equivalent to $4 ∶ 3$.

\begin{array}{rl} \underbrace{🍊🍊🍊🍊🍊🍊🍊🍊}&:;\underbrace{🍋🍋🍋🍋🍋🍋} \\\ 8;;;;;;;;;;;;;;;,,&;;;;;;;;;;;;;;;,6 \\\ \\\ \underbrace{🍊🍊🍊🍊}&:;\underbrace{🍋🍋🍋} \\\ 4;;;;;;;;&;;;;;;;;;,3 \end{array}

Similarly, the ratio of lemons (🍋🍋🍋🍋🍋🍋) to oranges (🍊🍊🍊🍊🍊🍊🍊🍊) is $6 ∶ 8$ (or $3 ∶ 4$).

\begin{array}{rl} 🍋🍋🍋🍋🍋🍋&:,🍊🍊🍊🍊🍊🍊🍊🍊 & \text{(6 : 8)}&&\qquad \\\ 🍋🍋🍋&:,🍊🍊🍊🍊 & \text{(3 : 4)}&&\qquad \end{array}

The ratio of oranges 🍊 to all the fruit in the basket is $8 ∶ 14$ (or $4 ∶ 7$).

\begin{array}{rl} 🍊🍊🍊🍊🍊🍊🍊🍊&:,🍋🍋🍋🍋🍋🍋🍊🍊🍊🍊🍊🍊🍊🍊 & \text{(8 : 14)}&&\qquad \end{array}

Notation for Ratios

Ratios can be written in several different form:

  • Words: “Three to one”
  • Words and numbers: “3 to 1”
  • With a colon: $3 : 1$
  • As a fraction: $\frac{3}{1}$

Scaling Ratios

Ratios can be scaled up or down by multiplying or dividing both numbers by the same amount.

\begin{array}{} \hline \text{Original ratio } &3:1 \\\ \hline \times \text{2} &6:2 \\\ \times \text{3} &9:3 \\\ \times \text{4} &12:4 \\\ \times \frac{1}{3} &1:\frac{1}{3} \\\ \end{array}

Mixing Orange Juice

Lin makes sparkling orange juice by mixing 3 liters of orange juice with 4 liters of soda water. Noah makes sparkling orange juice by mixing 4 liters of orange juice with 5 liters of soda.

  1. How do the two mixtures compare in taste? Consider drawing a double number line to represent the relationships.
Lin’s mixture tastes more like soda. Noah’s mixture tastes more like orange juice.
  1. How can Lin make her sparkling orange juice taste the same as Noah’s just by adding more of one ingredient? How much will she need?
If Lin adds \( \frac{1}{5} \) liter of orange juice, then the ratio of juice to sparkling water will be \( 16\text{:}20 \), which is equivalent to \( 3\frac{1}{5}\text{:}4 \) if you multiply by 5, which is equivalent to Noah’s ratio of \( 4 \text{:} 5 \).
  1. How can Noah make his sparkling orange juice taste the same as Lin’s just by adding more of one ingredient? How much will he need?
If Noah adds \( \frac{1}{3} \) liter of sparkling water, then the ratio of juice to sparkling water will be \( 4 \text{:} 5\frac{1}{3} \), which you can see is equivalent to \( 12 \text{:} 16 \) if you multiply by 3, which is equivalent to Lin’s ratio of \( 3 \text{:} 4 \).

How Many Pennies?

(Lesson 12, Practice Problems, Problem 1, Page 96)

Priya collected 2,400 grams of pennies in a fundraiser. Each penny has a mass of 2.5 grams. How much money did Priya raise? If you get stuck, consider using a table.

Number of penniesMass (grams)
12.5
?2,400
  1. Identify the known facts.
    • Total grams = 2,400
    • 1 penny = 2.5 grams
  2. Identify the question (“what”, “how many”, or “how much”).
    • “How much money did Priya raise?” In other words, “How many pennies does Preya have?”
  3. We can find out how many pennies there are by dividing the total grams by the weigh of one penny in grams. \[ 2,400 \div 2.5 = 960 \text{ pennies} \]
  4. We know that there are 100 pennies per US Dollar. Therefore, we can divide the total number pennies by 100. Dividing by 100 is as simple as moving the decimal point 2 places to the left. \[960 \div 100 = $9.60 \]

Videos

Math Antics: Ratios and Rates
Math Antics: Proportions
Unit 32: Introducing Ratios
Unit 3: Rates and Percentages
Source: https://class.ronliskey.com/study/mathematics-8/ratios/