Q40. - Algebra 1

Question

The table shows the percentage of households that have broadband Internet access and the average growth rates for two age groups. How long will it take for the percentages to be the same?

Source: Pew Internet B American Life Project

Step-by-Step Solution

Answer

The percentages of households that have broadband internet access for both age groups were the same in the year 2003, that is 3 years before 2006.

Step 1. State the formula.

The percentage of households that have broadband internet access after $x$ years is given as,

 

$P\left(1+\frac{GRP}{100}\right)$, where $P$ is the percentage of households that have broadband internet access in 2006 and $GRP$ is the Growth Rate Percentage for $x$ years.

Step 2. List the given data.

The table below shows the percentage of households that have broadband Internet access and the average growth rates for two age groups:

Step 3. Formulate the equation.

The growth rate percentage for $x$ years is nothing but the product of growth rate percentage per year and $x$.

 

Let the percentages be the same after $x$ years.

 

Then, the percentage of households that have broadband internet access after $x$ years for the two age groups are:

 

$52.5\left(1+\frac{42x}{100}\right)$ and $25.5\left(1+\frac{52x}{100}\right)$. It is given that these percentages are equal.

 

Translating the given condition into an equation,

 $52.5\left(1+\frac{42x}{100}\right)=25.5\left(1+\frac{52x}{100}\right)$

Step 4. Solve the obtained equation.

Solving,

 

$52.5\left(1+\frac{42x}{100}\right)=25.5\left(1+\frac{52x}{100}\right)$  (Given equation)

 

$52.5+\frac{2205x}{100}=25.5+\frac{1326x}{100}$  (Distributive property)

 

$52.5+\frac{2205x}{100}-\frac{1326x}{100}=25.5+\frac{1326x}{100}-\frac{1326x}{100}$  (Subtract $\frac{1326x}{100}$ from both sides)

 

$52.5+\frac{879x}{100}=25.5$  (Simplify)

 

$52.5+\frac{879x}{100}-52.5=25.5-52.5$  (Subtract $52.5$ from both sides)

 

$\frac{879x}{100}=-27$  (Simplify)

 

$\frac{879x}{100}\times \frac{100}{879}=-27\times \frac{100}{879}$  (Multiply both sides by $\frac{100}{879}$)

 

$x\approx -3.07$  (Simplify)

 

Rounding off to the nearest unit, $x\approx -3$

 

So, $x=-3$ is the required solution of the given equation.

Step 5. Interpret the obtained solution.

It was assumed that the percentages of households that have broadband internet access for both age groups are the same after $x$ years from the year 2006.

 

It was obtained that for the given condition to be satisfied, $x=-3$.

 

Then, the percentages were the same 3 years before 2006, that is, in the year 2003.