1.8 Ratio and Proportion

Key Concepts:

• Ratio:

  A ratio compares two or more quantities using division. It can be written as \( a : b \), which means \( \frac{a}{b} \).

• Simplifying Ratios:

  Ratios can be simplified by dividing all terms by their greatest common divisor (GCD).

• Proportion:

  Two quantities are said to be

  proportional

  if one quantity is a constant multiple of the other. In direct proportionality, as one quantity increases, the other increases at the same rate, and this relationship can be expressed as \( y \propto x \) or \( y = kx \), where \( k \) is the constant of proportionality. Conversely, two quantities are

  inversely proportional

  if one quantity increases while the other decreases such that their product remains constant. This relationship is expressed as \( y \propto \frac{1}{x} \) or \( y = \frac{k}{x} \), where \( k \) is the constant of proportionality.

• Cross Multiplication:

  Used to solve proportions: \[ \frac{a}{b} = \frac{c}{d} \Rightarrow a \cdot d = b \cdot c \]

• Speed, Distance, and Time:

  In proportional problems, use the relationship: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]

• Converting Between Fractions, Decimals, and Percentages:

  - Fraction to percentage: multiply by 100 - Decimal to percentage: multiply by 100 - Percentage to decimal: divide by 100

Example 1: Solve the ratio \(\frac{2}{5} = \frac{x}{15}\).

Step 1: Cross multiply the terms in the equation: \[ 2 \cdot 15 = 5 \cdot x. \] Step 2: Simplify the multiplication: \[ 30 = 5x. \] Step 3: Solve for x: \[ x = \frac{30}{5} = 6. \] Therefore, \(x = 6\).

Example 2: Three boys shared D 10,500.00 in the ratio 6:7:8. Find the largest share.

Step 1: Find the total ratio: \[ 6 + 7 + 8 = 21. \] Step 2: Divide the total amount by the total ratio to find the value of one part: \[ \text{Value of one part} = \frac{10,500}{21} = 500. \] Step 3: Find the largest share: \[ \text{Largest share} = 8 \times 500 = 4,000. \] Therefore, the largest share is D 4,000.

Example 3: Given that \(P \propto \frac{1}{\sqrt{r}}\) and \(P = 3\) when \(r = 16\), find the value of \(r\) when \(P = \frac{3}{2}\).

Solution: Step 1: Express \(P\) in terms of \(r\) using the proportionality constant \(k\): \[ P = \frac{k}{\sqrt{r}}. \] Step 2: Substitute \(P = 3\) and \(r = 16\) to find \(k\): \[ 3 = \frac{k}{\sqrt{16}}. \] \[ 3 = \frac{k}{4}. \] \[ k = 3 \times 4 = 12. \] Step 3: Use \(k = 12\) and \(P = \frac{3}{2}\) to find \(r\): \[ \frac{3}{2} = \frac{12}{\sqrt{r}}. \] \[ \sqrt{r} = \frac{12}{\frac{3}{2}}. \] \[ \sqrt{r} = \frac{12 \times 2}{3} = 8. \] Step 4: Square both sides to find \(r\): \[ r = 8^2 = 64. \] Therefore, the value of \(r\) when \(P = \frac{3}{2}\) is \(64\).