IB MYP 3 Mathematics 1.5 Ratio, Proportion and Percentage Study Notes - New Syllabus

1.5 – Ratio, Proportion and Percentage

Ratios, proportions and percentages are different ways of describing relationships between quantities. They are widely used to compare amounts, scale quantities, solve problems and describe changes.

Ratios

A ratio compares two or more quantities measured in the same units.

A ratio can be written using:

• a colon: \(3:5\)
• a fraction: \(\dfrac{3}{5}\)
• words: \(3\text{ to }5\)

For example, if a class contains \(12\) boys and \(18\) girls, the ratio of boys to girls is:

\(12:18\)

Divide both parts by their greatest common factor, \(6\):

\(12:18=2:3\)

Therefore, the simplest ratio of boys to girls is \(2:3\).

 Equivalent Ratios

Equivalent ratios represent the same relationship.

\(2:3=4:6=6:9=10:15\)

The ratio remains equivalent as long as both parts are multiplied or divided by the same number.

Example:

\(5:7\)

Multiplying both parts by \(4\):

\(5\times4:7\times4=20:28\)

Therefore:

\(5:7=20:28\)

Sharing a Quantity in a Given Ratio

To divide a quantity according to a ratio:

1. Add the parts of the ratio.
2. Divide the total quantity by the total number of parts.
3. Multiply by each ratio part.

For example, divide \(84\) in the ratio \(3:4\).

\(3+4=7\)

Each part represents:

\(84\div7=12\)

Therefore:

\(3\times12=36\)

\(4\times12=48\)

Answer: \(36:48\)

Check:

\(36+48=84\)

Proportion

A proportion states that two ratios or fractions are equal.

\(\dfrac{a}{b}=\dfrac{c}{d}\)

One useful method for solving a proportion is cross multiplication.

\(\dfrac{a}{b}=\dfrac{c}{d}\)

\(ad=bc\)

For example:

\(\dfrac{5}{8}=\dfrac{x}{24}\)

Cross multiply:

\(5\times24=8x\)

\(120=8x\)

\(x=15\)

Answer: \(x=15\)

Direct Proportion

Two quantities are in direct proportion when one quantity increases or decreases by a constant factor and the other changes by the same factor.

The relationship can be written as:

\(y=kx\)

where \(k\) is the constant of proportionality.

For example, if \(5\) notebooks cost \(\$20\), then the cost per notebook is:

\(20\div5=4\)

So the cost is directly proportional to the number of notebooks:

\(C=4n\)

Therefore, \(8\) notebooks cost:

\(C=4(8)=32\)

Answer: \(\$32\)

Percentages

A percentage represents a quantity out of \(100\).

\(1\%=\dfrac{1}{100}=0.01\)

The three equivalent forms are:

Converting Percentages into Decimals and Fractions

A percentage means “out of \(100\)”. Therefore, a percentage can be converted into a fraction by placing it over \(100\), and into a decimal by dividing by \(100\).

Percentage → Fraction

\(35\%=\dfrac{35}{100}=\dfrac{7}{20}\)

Percentage → Decimal

\(35\%=0.35\)

Converting Decimals and Fractions into Percentages

Decimal → Percentage

Multiply the decimal by \(100\%\).

\(0.42\times100\%=42\%\)

Fraction → Percentage

Convert the fraction to a decimal or multiply by \(100\%\).

\(\dfrac{3}{5}\times100\%=60\%\)

Example:

\(\dfrac{7}{8}\times100\%=87.5\%\)

The Unitary Method for Percentages

The unitary method finds the value of \(1\%\) first and then uses it to find the required percentage.

For example, find \(18\%\) of \(450\).

First find \(1\%\):

\(450\div100=4.5\)

Then find \(18\%\):

\(4.5\times18=81\)

Answer: \(81\)

Finding a Percentage Change

Percentage change measures how much a quantity has changed compared with its original value.

\(\text{Percentage change}=\dfrac{|\text{New value}-\text{Original value}|}{\text{Original value}}\times100\%\)

• If the new value is greater, the result is a percentage increase.
• If the new value is smaller, the result is a percentage decrease.

For example, a quantity changes from \(80\) to \(92\).

\(\text{Change}=92-80=12\)

\(\text{Percentage change}=\dfrac{12}{80}\times100\%=15\%\)

Answer: \(15\%\) increase.

Finding the Original Amount

When the final amount and percentage change are known, the original amount can be found using the percentage multiplier.

After an increase of \(p\%\):

\(\text{Original}=\dfrac{\text{Final}}{1+\frac{p}{100}}\)

After a decrease of \(p\%\):

\(\text{Original}=\dfrac{\text{Final}}{1-\frac{p}{100}}\)

Example: A jacket costs \(\$144\) after a \(20\%\) discount. Find its original price.

\(\text{Multiplier}=1-0.20=0.80\)

\(\text{Original}=\dfrac{144}{0.80}=\$180\)

Answer: \(\$180\)

Profit and Loss

When an item is bought and then sold, the difference between the selling price and cost price determines whether there is a profit or loss.

Profit:

\(\text{Profit}=\text{Selling Price}-\text{Cost Price}\)

Loss:

\(\text{Loss}=\text{Cost Price}-\text{Selling Price}\)

Profit Percentage:

\(\text{Profit\%}=\dfrac{\text{Profit}}{\text{Cost Price}}\times100\%\)

Loss Percentage:

\(\text{Loss\%}=\dfrac{\text{Loss}}{\text{Cost Price}}\times100\%\)

Example: An item is bought for \(\$240\) and sold for \(\$300\).

\(\text{Profit}=300-240=60\)

\(\text{Profit\%}=\dfrac{60}{240}\times100\%=25\%\)

Answer: \(25\%\) profit.

Discount

A discount is a reduction in the marked or original price of an item.

Discount amount:

\(\text{Discount}=\dfrac{\text{Discount\%}}{100}\times\text{Original Price}\)

Sale price:

\(\text{Sale Price}=\text{Original Price}-\text{Discount}\)

Alternatively:

\(\text{Sale Price}=\text{Original Price}\times\left(1-\dfrac{\text{Discount\%}}{100}\right)\)

Example: A phone costs \(\$800\) and is discounted by \(15\%\).

\(\text{Discount}=0.15\times800=\$120\)

\(\text{Sale Price}=800-120=\$680\)

Answer: \(\$680\)

VAT and GST

VAT (Value Added Tax) and GST (Goods and Services Tax) are taxes added to the price of goods or services. The calculation is the same percentage method; the tax rate depends on the situation.

Tax amount:

\(\text{Tax}=\dfrac{\text{Tax Rate}}{100}\times\text{Price}\)

Price including tax:

\(\text{Final Price}=\text{Price}+\text{Tax}\)

Or using a multiplier:

\(\text{Final Price}=\text{Price}\times\left(1+\dfrac{\text{Tax Rate}}{100}\right)\)

Example: A service costs \(\$500\) before tax and the applicable tax rate is \(10\%\).

\(\text{Tax}=0.10\times500=\$50\)

\(\text{Final Price}=500+50=\$550\)

Answer: \(\$550\)

Finding a Percentage of a Quantity

To find a percentage of a quantity, convert the percentage to a decimal and multiply.

\(\text{Percentage of quantity}=\dfrac{p}{100}\times\text{quantity}\)

For example, find \(35\%\) of \(240\).

\(35\%=\dfrac{35}{100}=0.35\)

\(0.35\times240=84\)

Answer: \(84\)

 Finding What Percentage One Quantity Is of Another

To find what percentage one quantity represents of another:

\(\text{Percentage}=\dfrac{\text{part}}{\text{whole}}\times100\%\)

For example, \(18\) students out of \(30\) completed a task.

\(\dfrac{18}{30}\times100\%=60\%\)

Therefore, \(60\%\) of the students completed the task.

 Reverse Percentages

Sometimes the final value is known and the original value must be found.

If a quantity has increased by \(20\%\), the final value is \(120\%\) of the original:

\(\text{Final value}=1.20\times\text{Original value}\)

Therefore:

\(\text{Original value}=\dfrac{\text{Final value}}{1.20}\)

For example, after a \(20\%\) increase, a price is \(\$180\). Find the original price.

\(\text{Original}=\dfrac{180}{1.20}=150\)

Answer: \(\$150\)