How do we practically use ratios?

A ratio is characterised as a tool used to analyse the size of at least two values with respect to each other.

A ratio is a fraction where the numerator is implied as a predecessor, and the denominator is implied as to the following. We use a colon image $$:$$ to form a ratio. For example, $$3:4$$, $$1:3$$, $$5:7$$, $$1:1$$, and so on are instances of ratios.

E1.11: Increase and decrease a quantity by a given ratio

When the ratio of the new amount to an old amount is in the form of an improper fraction, at that point, the new amount is greater than the old amount. Applying this ratio to the old amount is known to be increasing the old amount in a given ratio. 

When the ratio of the new amount to an old amount is in the form of a proper fraction, at that point the new amount is less than the old amount. Applying this ratio to the old amount is known as decreasing the old amount in a given ratio.

For instance, a car driver is driving a car on a highway. His average speed is $$80\;\text{kmph}$$. He has to travel a distance of $$240\;\text{kms}$$ on this highway. At this speed, he will cover this distance of $$240 \;\text{kms}$$ in exactly $$3\;\text{hours}$$. Due to some urgent work, he has to cover this distance in $$2 \;\text{hours}$$, and to cover this distance of $$240 \;\text{kms}$$, he has to travel at a speed of $$120 \;\text{kmph}$$. So, in this scenario, he has to increase his speed in the ratio $$120:80$$, which can be written in its simplest form $$3:2$$. It is how we use a ratio to show increase and decrease in everything.

Worked examples

Example 1: Change $$50$$ in the ratio of $$7:5$$.

Step 1: Change the given ratio in the form of a fraction

$$7:5=\frac{7}{5}$$

Step 2: Check whether the quantity would decrease or increase after changing 

The above form of fraction makes an improper fraction, meaning that the quantity increases after the change.

Step 3: Multiply the old quantity with the above-formed fraction to find the new quantity

 $$50\times \frac{7}{5}$$

Step 4: Write the final answer.

The new number after changing $$50$$ in the ratio $$7:5$$ is $$70$$.

Example 2: Paul had been working in a company for $$30 \;\text{years}$$. After working for $$30 \;\text{years}$$, he got retirement funds of value $$\$100000$$, which he had saved in a scheme of retirement investment savings. Due to the global financial crisis, his savings decreased to the ratio of $$6:7$$. If he had savings of $$\$70000$$ in his savings scheme last year, what is the value of the amount he has left this year?

Step 1: Change the given ratio in the form of a fraction

$$6:7=\frac{6}{7}$$

Step 2: Check whether the quantity decreases or increases after changing it in the given ratio

Since the above fraction is a proper fraction, the value of the quantity decreases.

Step 3: Multiply the old quantity with the ratio to find the new quantity

$$7000\times \frac{6}{7}$$

Step 4: Write the final answer.

Thus, Paul had $$\$60000$$ left this year after decreasing his savings due to the global financial crisis.