Ratio and Proportion for ESAT Mathematics

Simple Proportion

Simple proportion problems involve finding a new quantity based on a known relationship between two values. If it is known that $x$ items cost $\mathcal{E}y$, we can determine the cost of any other number of items, $a$, using two primary techniques.

Method 1: The Unitary Method

In the unitary method, you first find the value of a single unit. Consider the following example: If 12 boxes of biscuits cost 42, how much do 20 boxes cost?

1. Find the cost of one box: 1 box costs $42 \div 12$. It is best not to divide this out at this stage unless it results in an integer, to avoid rounding errors.

2. Multiply by the required number: 20 boxes cost $42 \div 12 \times 20$.

3. Simplify the calculation: $42 \div 12 \times 20 = \frac{42 \times 20}{12}$. Dividing the numerator and denominator by 6 gives $\frac{7 \times 20}{2}$. Further dividing by 2 gives $\frac{7 \times 10}{1} = 70$. Thus, 20 boxes cost 70.

Method 2: Using the Highest Common Factor (HCF)

Alternatively, you can use the Highest Common Factor of the given and required quantities to simplify the arithmetic. Using the same example of 12 and 20 boxes:

1. The HCF of 12 and 20 is 4. Therefore, calculate the cost of 4 boxes.

2. Since 12 boxes cost 42, 4 boxes cost one third of that: $42 \div 3 = 14$.

3. Since 20 boxes is five times 4 boxes, multiply the cost by 5: $14 \times 5 = 70$. This confirms the cost of 20 boxes is 70.

Relating Ratios to Fractions

If a mixture consists only of two components $x$ and $y$ in the ratio $a:b$, the fraction of the total mixture made up by $x$ is $\frac{a}{a+b}$. When dealing with multiple ratios, you must find a common link to combine them into a single ratio string.

Example: A packet of sweets contains yellow, green, and red sweets. The ratio of yellow to green is $3:4$ and the ratio of green to red is $6:9$. To find what fraction of the pack is yellow, we must find the combined ratio $yellow:green:red$.

1. Identify the common component, which is green sweets. In the first ratio, green is 4, and in the second, it is 6.

2. Express $3:4$ in terms of 6 by multiplying by $\frac{3}{2}$ or 1.5. This gives $3 \times 1.5 : 4 \times 1.5$, which is $4.5:6$.

3. The combined ratio is $4.5:6:9$. To work with integers, multiply through by 2 to get $9:12:18$.

4. Calculate the fraction of yellow sweets: $\frac{9}{9+12+18} = \frac{9}{39}$. Simplified, this is $\frac{3}{13}$.

Relating Ratios to Linear Functions

Ratios can also be expressed as algebraic relationships between two variables. If the ratio $x:y$ is $a:b$, then the relationship is $bx = ay$ or $y = \frac{b}{a}x$.

Consider the ratio $x:y = 2:3$. To write $y$ as a linear function of $x$:

1. Convert the ratio into a fractional equation: $\frac{x}{y} = \frac{2}{3}$.

2. Multiply both sides by $y$: $x = \frac{2}{3}y$.

3. Multiply both sides by 3 to remove the denominator: $3x = 2y$.

4. Divide both sides by 2 to isolate $y$: $y = \frac{3x}{2}$, or $y = 1.5x$. This shows that $y$ is a linear function of $x$ passing through the origin.