Dividing Quantities in Given Ratios Revision Notes for VCE SSCE General Mathematics

Dividing Quantities in Given Ratios

Understanding the concept

When you divide a quantity according to a ratio, you're splitting it into parts based on the numbers in that ratio. The key is to think of the ratio as representing portions or shares.

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For example, if a quantity is divided in the ratio $5:3$ , there are $5 + 3 = 8$ parts in total. After division, one share has 5 parts and the other has 3 parts. If you divide $8 between two people in the ratio $5:3$ , one person gets $5 and the other gets $3.

The step-by-step method

To divide any quantity in a given ratio, follow these four steps:

Step 1: Add up all the numbers in the ratio to find the total number of parts.

Step 2: Divide the total quantity by the total number of parts. This tells you the value of one part.

Step 3: Multiply the value of one part by each number in the ratio to find each share.

Step 4: Check your answer by adding all shares together. They should equal the original quantity.

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Common Mistake to Avoid:

Many students forget to check their work at the end. Always verify that your calculated shares add up to the original total quantity. If they don't match, you've made an error somewhere in your calculations.

Worked examples

Let's see how this method works with different ratios.

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Worked Example: Dividing 60 students in the ratio 1:3

Calculate the number of students in each class if 60 students are divided in the ratio $1:3$ .

Step 1: Find the total number of parts

The total number of parts is $1 + 3 = 4$ .

Step 2: Calculate the value of one part

$60 \div 4 = 15$

One group of students will have $1 \times 15 = 15$ students.

Step 3: Calculate the other share

The other group will have $3 \times 15 = 45$ students.

Step 4: Check the total

$15 + 45 = 60 \text{ } ✓$

Answer: The two groups will have 15 and 45 students.

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Worked Example: Dividing 60 students in the ratio 1:5

Calculate the number of students in each class if 60 students are divided in the ratio $1:5$ .

Step 1: Find the total number of parts

The total number of parts is $1 + 5 = 6$ .

Step 2: Calculate the value of one part

$60 \div 6 = 10$

One group of students will have $1 \times 10 = 10$ students.

Step 3: Calculate the other share

The other group will have $5 \times 10 = 50$ students.

Step 4: Check the total

$10 + 50 = 60 \text{ } ✓$

Answer: The two groups will have 10 and 50 students.

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Worked Example: Dividing 60 students in the ratio 1:2:7

Calculate the number of students in each class if 60 students are divided in the ratio $1:2:7$ .

Notice this ratio has three parts instead of two. The method is exactly the same!

Step 1: Find the total number of parts

The total number of parts is $1 + 2 + 7 = 10$ .

Step 2: Calculate the value of one part

$60 \div 10 = 6$

One group of students will have $1 \times 6 = 6$ students.

Step 3: Calculate the other shares

The other groups will have $2 \times 6 = 12$ students and $7 \times 6 = 42$ students.

Step 4: Check the total

$6 + 12 + 42 = 60 \text{ } ✓$

Answer: The three groups will have 6, 12 and 42 students.

Practice question

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Practice Question:

Divide $36 in the ratio $1:2:3$ .

Hint 1: Add up the total number of parts.

Hint 2: Divide $36 by this total to give the amount of one part.

Hint 3: Multiply this amount by each quantity in the ratio.

Real-world application: recipe scaling

Ratios are useful when scaling recipes. Here's an example:

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The recipe shown makes 25 Anzac biscuits. If you wanted to make 75 biscuits (which is 3 times as many), you would need to multiply each ingredient quantity by 3.

The ratio of rolled oats to coconut to flour to brown sugar to butter in the original recipe is:

$100:60:175:125:125 \text{ (in grams)}$

Key formula

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Essential Formula for Dividing Quantities in Ratios

To divide a quantity in a given ratio $a:b$ :

Divide the quantity by the sum of $a$ and $b$
This gives the allocation of one part
Multiply this one part value by $a$ and then by $b$ to give the required allocations

For three-part ratios $a\:b:c$ , the process is the same but you divide by $a + b + c$ and multiply the one part value by each of $a$ , $b$ , and $c$ .

Remember!

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Key Points to Remember:

Find total parts first: Always add all numbers in the ratio before doing anything else.
One part is the key: Dividing the total quantity by the total parts gives you the value of one part.
Multiply to allocate: Multiply the one part value by each number in the ratio to find each share.
Always check: Your final shares should add up to the original quantity.
Works for any number of parts: The method works the same whether you're dividing in 2, 3, or more parts.

Dividing Quantities in Given Ratios (VCE SSCE General Mathematics): Revision Notes