Question

Samantha is \(1.50 \mathrm{m}\) tall on her eleventh birthday and $1.65 \mathrm{m}$ tall on her twelfth birthday. By what factor has her height increased? By what percentage?

Short answer

Answer: Samantha's height increased by a factor of 1.1 and the percentage increase was 10%.

Step-by-step solution

Calculate the absolute increase in height

Samantha's height increased from \(1.50\,\text{m}\) to \(1.65\,\text{m}\). To find the absolute increase in her height, we need to subtract her height at 11 years old from her height at 12 years old: $$1.65\,\text{m} - 1.50\,\text{m} = 0.15\,\text{m}$$ So, Samantha's height increased by \(0.15\,\text{m}\).

Calculate the factor of increase

The factor of increase is the ratio of the final height to the initial height. We can find this by dividing Samantha's height at 12 years old by her height at 11 years old: $$\frac{1.65\,\text{m}}{1.50\,\text{m}} = 1.1$$ So, Samantha's height increased by a factor of 1.1.

Calculate the percentage increase

To find the percentage increase, we will use the formula: $$\text{percentage increase} = \left(\frac{\text{final height} - \text{initial height}}{\text{initial height}}\right) \times 100$$ Plugging in the values, we have: $$\text{percentage increase} = \left(\frac{1.65\,\text{m} - 1.50\,\text{m}}{1.50\,\text{m}}\right) \times 100$$ $$\text{percentage increase} = \left(\frac{0.15\,\text{m}}{1.50\,\text{m}}\right) \times 100$$ $$\text{percentage increase} = 0.1 \times 100$$ $$\text{percentage increase} = 10\%$$ So, Samantha's height increased by 10% from her 11th to her 12th birthday.