Ratio Problem Solving

 

Ratios & Fractions, Decimals, and Percentages (FDP)

How Are Ratios and Fractions Linked?

• A fraction compares a part to the whole.
• A ratio compares one part to another part.

Example: Sharing a Pizza

A pizza is sliced into 8 pieces and shared between two people.

• Person 1 gets 5 slices
• Person 2 gets 3 slices

As a fraction of the whole pizza:

• Person 1 receives $\frac{5}{8}$​
• Person 2 receives $\frac{3}{8}$​

As a ratio of slices:

• The ratio of Person 1:Person 2 is 5:3
• The total number of parts in the ratio is $5 + 3 = 8$

Unlike ratios, fractions can be converted into percentages or decimals.

 

Solving Ratio Problems with FDP

Some ratio problems involve fractions or percentages.

 

Example 1: Finding a Fraction of a Fraction

In a school, $\frac{3}{40}$​ of the people are staff. Of those, $\frac{2}{3}$​ wore Christmas jumpers.

To find the fraction of all people wearing jumpers:

$$\frac{3}{40} \times \frac{2}{3} = \frac{6}{120} = \frac{1}{20}$$

Final Answer: $\frac{1}{20}$ of the school wore Christmas jumpers.

 

Example 2: Finding a Percentage of a Fraction

In the same school, $\frac{37}{40}$​ are students. Of those, $88\%$ wore Christmas jumpers.

$$\frac{37}{40} \times 0.88 = 0.814$$

Final Answer: $81.4\%$ of the school were students wearing jumpers.

 

 

 

 

 

Combining Two Ratios into a Three-Part Ratio

Example: Animals on a Farm

A farm has 85 animals, with the following ratios:

• Cows:Sheep = 2:3
• Sheep:Pigs = 6:7

Find the number of cows, sheep, and pigs.

 

Step 1: Find a Common Value for Sheep

We have:

$$C:S = 2:3, \quad S:P = 6:7$$

Since sheep appears in both, we need a common value.

Multiply the C:S ratio by 2 so that sheep = 6 in both ratios:

$$C:S = 4:6, \quad S:P = 6:7$$

Now, we can combine them into a three-part ratio:

$$C:S:P = 4:6:7$$

Step 2: Find the Total Number of Parts

$$4 + 6 + 7 = 17$$

Step 3: Find the Value of One Part

$$85 \div 17 = 5$$

Step 4: Multiply Each Part by 5

$$Cows = 4 \times 5 = 20, \quad Sheep = 6 \times 5 = 30, \quad Pigs = 7 \times 5 = 35$$

Final Answer: 20 cows, 30 sheep, and 35 pigs.