Calculations with Ratios Revision Notes for HSC SSCE Mathematics Standard

Calculations with Ratios

Understanding ratios

A ratio is a way to compare amounts of the same type in a specific order. When we write a ratio like $3:4$ , we're saying that for every $3$ parts of one quantity, there are $4$ parts of another quantity.

Ratios can be expressed in several different ways:

As a ratio: $3:4$
As a fraction: $\frac{3}{4}$
As a decimal: $0.75$
As a percentage: $75\%$

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You see ratios in everyday life. For example, television screens and computer monitors often have an aspect ratio of $16:9$ , which describes the relationship between width and height.

Equivalent ratios

Just like fractions, ratios can be simplified or expanded while maintaining the same relationship. We create equivalent ratios by multiplying or dividing both amounts in the ratio by the same number.

In the diagram above, the ratio $15:12$ can be simplified to $5:4$ by dividing both numbers by $3$ (the highest common factor). Similarly, we can expand $5:4$ back to $15:12$ by multiplying both numbers by $3$ . These ratios are equivalent because they represent the same relationship.

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Key principle: Equivalent ratios are obtained by multiplying or dividing each amount by the same number.

Simplifying ratios

To simplify a ratio means to express it in its lowest terms, similar to simplifying a fraction. The method you use depends on what type of numbers are in the ratio.

Simplifying ratios with fractions

When a ratio contains fractions, multiply both sides by the lowest common denominator (LCD) to eliminate the fractions.

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Worked Example: Simplifying a ratio with fractions

Simplify $3:\frac{1}{2}$

Step 1: Identify that we need to multiply both sides by $2$ (the denominator)

$3:\frac{1}{2} = 3 \times 2:\frac{1}{2} \times 2$

Step 2: Calculate the result

$= 6:1$

Answer: The simplified ratio is 6:1.

Simplifying ratios with decimals

When a ratio contains decimals, multiply both sides by a power of 10 to convert to whole numbers, then simplify by dividing by the highest common factor.

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Worked Example: Simplifying a ratio with decimals

Simplify $1.5:3.5$

Step 1: Multiply both sides by $10$ to remove decimals

$1.5:3.5 = 1.5 \times 10:3.5 \times 10$

Step 2: Calculate

$= 15:35$

Step 3: Find the highest common factor of $15$ and $35$ , which is $5$

Step 4: Divide both sides by $5$

$= \frac{15}{5}:\frac{35}{5}$

Step 5: Calculate the final answer

$= 3:7$

Answer: The simplified ratio is 3:7.

Simplifying ratios with fractions only

When both parts of a ratio are fractions, convert them to a common denominator, then use the numerators to form the simplified ratio.

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Worked Example: Simplifying a ratio with two fractions

Simplify $\frac{2}{7}:\frac{5}{21}$

Step 1: Find a common denominator. The LCD of $7$ and $21$ is $21$

Step 2: Convert both fractions to have denominator $21$

$\frac{2}{7}:\frac{5}{21} = \frac{6}{21}:\frac{5}{21}$

Step 3: Since the denominators are the same, use the numerators as the ratio

$= 6:5$

Answer: The simplified ratio is 6:5.

Dividing a quantity in a given ratio

Sometimes you need to split a total amount according to a given ratio. This is common in sharing problems, recipe scaling, or dividing money or resources.

Method for dividing quantities

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Follow these four steps to divide a quantity in a given ratio:

Calculate the total number of parts by adding each amount in the ratio together
Divide the quantity by the total number of parts to find the value of one part
Multiply each amount in the ratio by the value of one part
Check your answer by adding all the parts together - they should equal the original quantity

Two-part ratio example

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Worked Example: Dividing money in a two-part ratio

Problem: Mikhail and Ilya were given $450 to share in the ratio $4:5$ . How much did each person receive?

Solution:

Step 1: Calculate the total number of parts

$\text{Total parts} = 4 + 5 = 9$

Step 2: Find the value of one part

$\text{9 parts} = \$450$

$\text{1 part} = \frac{\$450}{9} = \$50$

Step 3: Calculate each person's share

Mikhail's share: $4 \times 50 = 200$
Ilya's share: $5 \times 50 = 250$

Step 4: Check the answer

$\$200 + \$250 = \$450$ ✓

Answer: Mikhail received $200 and Ilya received $250.

Three-part ratio example

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Worked Example: Dividing money in a three-part ratio

Problem: A man left $6000 to be divided among his three children, Xia, Yui and Zi, in the ratio $5:8:7$ in that order. How much did each child receive?

Solution:

Step 1: Calculate the total number of parts

$\text{Total parts} = 5 + 8 + 7 = 20$

Step 2: Find the value of one part

$\text{20 parts} = \$6000$

$\text{1 part} = \frac{\$6000}{20} = \$300$

Step 3: Calculate each child's share

Xia's share: $5 \times 300 = 1500$
Yui's share: $8 \times 300 = 2400$
Zi's share: $7 \times 300 = 2100$

Step 4: Check the answer

$\$1500 + \$2400 + \$2100 = \$6000$ ✓

Answer: Xia received $1500, Yui received $2400, and Zi received $2100.

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Exam tip: Always check your answer by adding all the parts back together. This quick verification can catch calculation errors before you submit your work.

Remember!

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Key Points to Remember:

A ratio compares amounts of the same units in a specific order
Ratios can be expressed as fractions, decimals, or percentages
Create equivalent ratios by multiplying or dividing both amounts by the same number
To simplify ratios with fractions, multiply by the lowest common denominator
To simplify ratios with decimals, multiply by a power of 10, then divide by the highest common factor
When dividing a quantity in a ratio, use the four-step method: calculate total parts, find one part, multiply for each share, and check your answer

Calculations with Ratios (HSC SSCE Mathematics Standard): Revision Notes