11+ Maths

Ratio and Proportions

Objective: Gain a strong understanding, recognize and express the quantitative relationship between two or more quantities.

Why Ratio and Proportions Matters: Ratio allow us to compare quantities effectively, such as understanding speed or density. Proportions are used in tasks like creating models, scaling recipes, mixing solutions or calculating discounts and interest rates.

Key Ideas:

Ratio expresses a relationship between two quantities, showing how many times, one value contains or is contained within the other.
Proportion represents the equality of two ratios.

Introduction to Ratios

What is ratio? A ratio is a relationship between two numbers showing how many times the first number contains the second.
Symbolism:
- $Ratio=ab3\text{Ratio} = \frac{a}{b}$ $a : b$ , where $a$ and $b$ are two number $.$
Example: The ratio of boys to girls in a class of 2 boys and 4 girls is 2:4, which simplifies to 1:2.

\frac{3}{4}

Types of Ratios

\frac{pears}{oranges}

Part-to-Part Ratios:
- Example: Ratio of pears to oranges in a basket.
- Formula: $applesoranges\frac{\text{apples}}{\text{oranges}}distancetime\frac{\text{distance}}{\text{time}}$

Ratio =

\frac{pears}{oranges}

Types of Ratios

\frac{pears}{oranges}

Part-to-Part Ratios:
- Example: Ratio of pears to oranges in a basket.
- Formula: $applesoranges\frac{\text{apples}}{\text{oranges}}distancetime\frac{\text{distance}}{\text{time}}$

Ratio =

\frac{6}{oranges}

Types of Ratios

\frac{pears}{oranges}

Part-to-Part Ratios:
- Example: Ratio of pears to oranges in a basket.
- Formula: $applesoranges\frac{\text{apples}}{\text{oranges}}distancetime\frac{\text{distance}}{\text{time}}$

Ratio =

\frac{6}{9}

or Ratio= 6 : 9

Simplifying Ratios

Ratio =

\frac{6}{9}

Ratio can be simplified, the same way as fractions. To simplify a ratio, divide all parts of the ratio by their greatest common factor (GCF). Always simplify ratios to their smallest whole numbers.

\frac{2}{3}

Example:

The GCF (6 ,9) = 3

You can notice that every 2 pears correspond to 3 oranges.

Ratio =

\frac{6}{9}

\frac{2}{3}

Simplifying Ratios

Ratio can be simplified, the same way as fractions. To simplify a ratio, divide all parts of the ratio by their greatest common factor (GCF). Always simplify ratios to their smallest whole numbers.

Types of Ratios

\frac{basketball \ balls}{total \ balls}

Part-to-Whole Ratios:
- Example: Ratio of basketball balls to total balls.
- Formula: $marbles\frac{\text{red marbles}}{\text{total marbles}$

Quick Check:

Find the ratio of basketball balls to the total number of balls

Types of Ratios

\frac{basketball \ balls}{total \ balls}

Part-to-Whole Ratios:
- Example: Ratio of basketball balls to total balls.
- Formula: $marbles\frac{\text{red marbles}}{\text{total marbles}$

Quick Check:

Find the ratio of basketball balls to the total number of balls

Answer:

Ratio =

\frac{6}{14}

or Ratio = 6 : 14

Types of Ratios

\frac{basketball \ balls}{total \ balls}

Part-to-Whole Ratios:
- Example: Ratio of basketball balls to total balls.
- Formula: $marbles\frac{\text{red marbles}}{\text{total marbles}$

Quick Check:

Find the ratio of basketball balls to the total number of balls

Answer:

Ratio =

\frac{6}{14}

\frac{3}{7}

or Ratio = 3 : 7

Do not forget to simplify it.

GCF (6,14) = 2

Introduction to Proportions

A proportion is an equation that expresses two ratios as equal.
A proportion can be written as:

\frac{a}{b} = \frac{c}{d}

where $a, b, c,$ and $d$ are numbers.

Example: If we consider one half of a full circle, we can write it down as 1:2, which equals to 2:4, or 4:8 and so on. We can write the proportion as:

Solving Proportions

To solve a proportion we use the cross-multiplication method.

\frac{a}{b} = \frac{c}{d}

where $a, b, c,$ and $d$ are numbers.

Example: If 4 oranges cost £2, how much will 10 oranges cost at the same rate?

a \times b = c \times d

Answer: The proportion can be set up as:

Where:

$4$ is the number of oranges,
$2$ is the cost for 4 oranges,
$10$ is the number of oranges for which we need to find the cost,
$x$ is the unknown cost of 10 oranges.

\frac{a}{b} = \frac{c}{d}

\frac{4}{10} = \frac{2}{x}

Solving Proportions

To solve a proportion we use the cross-multiplication method.

\frac{a}{b} = \frac{c}{d}

where $a, b, c,$ and $d$ are numbers.

Example: If 4 oranges cost £2, how much will 10 oranges cost at the same rate?

a \times b = c \times d

Answer: The proportion can be set up as:

Where:

$4$ is the number of oranges,
$2$ is the cost for 4 oranges,
$10$ is the number of oranges for which we need to find the cost,
$x$ is the unknown cost of 10 oranges.

\frac{a}{b} = \frac{c}{d}

\frac{4}{10} = \frac{2}{x}

4 oranges cost £2

10 oranges cost £x

Solving Proportions

To solve a proportion we use the cross-multiplication method.

\frac{a}{b} = \frac{c}{d}

where $a, b, c,$ and $d$ are numbers.

Example: If 4 oranges cost £2, how much will 10 oranges cost at the same rate?

a \times b = c \times d

\frac{4}{10} = \frac{2}{x}

Answer: Using the cross-multiplication method we get:

4 \times x = 2 \times 10

4x = 20

By dividing the both sides of the equation by 4 we get:

x = 5

We get that the price for 10 oranges is £5.

Application of Ratio and Proportions

Real-World Examples:

Recipe Scaling: If a recipe calls for 2 cups of flour for 4 servings, how much flour is needed for 10 servings?
Map Reading: If 1 cm on a map equals 100 km, how many kilometres does 5 cm represent?
Medication Dosages: A doctor prescribes 10 mg of medicine per 5 kg of body weight. How much medicine should a 70 kg patient take?
Population Ratios: In a city of 100,000 people, 60% are adults. How many adults live in the city?

$1100=3x\frac{1}{100} = \fr$

Application of Ratio and Proportions

Real-World Examples:

Recipe Scaling: If a recipe calls for 2 cups of flour for 4 servings, how much flour is needed for 10 servings?
Map Reading: If 1 cm on a map equals 100 km, how many kilometres does 5 cm represent?
Medication Dosages: A doctor prescribes 10 mg of medicine per 5 kg of body weight. How much medicine should a 70 kg patient take?
Population Ratios: In a city of 100,000 people, 60% are adults. How many adults live in the city?

$1100=3x\frac{1}{100} = \fr$

Before we review the solution, try working through these problems step-by-step. You'll be surprised how simple it is!

Application of Ratio and Proportions

Real-World Examples:

1. Recipe Scaling: If a recipe calls for 3 cups of flour for 6 servings, how much flour is needed for 10 servings?

Answer: We need 5 cups of flour for 10 servings.

2. Map Reading: If 1 cm on a map equals 100 km, how many kilometres does 5 cm represent?

$1100=3x\frac{1}{100} = \fr$

\frac{1}{5} = \frac{100}{x}

1 \times x = 5 \times 100

x = 500

Answer: 5cm on the map represents 500km in real life.

\frac{3}{x} = \frac{6}{10}

3 \times 10 = 6 \times x

30 = 6x

x = 5

Application of Ratio and Proportions

Real-World Examples:

3. Medication Dosages: A doctor prescribes 10 mg of medicine per 5 kg of body weight. How much medicine should a 70 kg patient take?

4. Population Ratios: In a city of 100,000 people, 60% are adults. How many adults live in the city?

$1100=3x\frac{1}{100} = \fr$

\frac{60}{100} = \frac{x}{100000}

100 \times x = 60 \times 100000

100x = 6000000

Answer: In the city live 60,000 adults.

\frac{10}{x} = \frac{5}{70}

10 \times 70 = 5 \times x

700 = 5x

Answer: 70kg person should take 140mg of medicine.

x = 140

x = 60000

What We Learned About Ratios and Proportions

Summary: Ratios compare quantities, and proportions show the equality of two ratios. Mastering ratios and proportions helps with real-world problems like scaling, map reading, and recipe adjustments.
Proportions help us solve everyday problems, like scaling recipes, comparing prices, or working out distances on a map.
Ratios can also help us make decisions, such as adjusting ingredients for a larger or smaller batch of something or figuring out the most efficient way to mix materials.
Remember: Ratios and proportions are not just mathematical tools—they are ways of thinking critically and solving problems that we encounter daily!