SAT Percent Change Practice

It's given that in 2008 Zinah earned 14% more than in 2007. Let $h$ represent the amount Zinah earned in 2007 and let $j$ represent the amount that Zinah earned in 2008.

This situation can be represented by the equation $j=1+\frac{14}{100}h$, or $j=1.14h$. It's also given that in 2009 Zinah earned 4% more than in 2008.

Let $k$ represent the amount Zinah earned in 2009. This situation can be represented by the equation $k=1+\frac{4}{100}j$, or $k=1.04j$.

Substituting $1.14h$ for $j$ in the equation $k=1.04j$ yields $k=1.04\times 1.14h$, or $k=1.1856h$. If Zinah earned $y$ times as much in 2009 as in 2007, then the value of $y$ is 1.1856.

It's given that the value of the comic book increased by 167% from the end of 2011 to the end of 2012. Therefore, if the value of the comic book at the end of 2011 was $x$ dollars, then the value, in dollars, of the comic book at the end of 2012 was $x+\left(\frac{167}{100}\right)x$, which can be rewritten as $1x+1.67x$, or $2.67x$.

It's also given that the value of the comic book decreased by 16% from the end of 2012 to the end of 2013. Therefore, the value, in dollars, of the comic book at the end of 2013 was $2.67x-2.67x\left(\frac{16}{100}\right)$, which can be rewritten as $2.67x-(2.67x)(0.16)$, or $2.2428x$.

Thus, if the value of the comic book at the end of 2011 was $x$ dollars, and the value of the comic book at the end of 2013 was $2.2428x$ dollars, then from the end of 2011 to the end of 2013, the value of the comic book increased by $2.2428x-1x$, or $1.2428x$, dollars. Therefore, the increase in the value of the comic book from the end of 2011 to the end of 2013 is equal to 1.2428 times the value of the comic book at the end of 2011.

It follows that from the end of 2011 to the end of 2013, the net percentage increase in the value of the comic book was $(1.2428)(100)\%$, or 124.28%.

It's given that the number of coins in the collection increased from 9 to 90. It follows that the number of coins in the collection increased by $90-9$, or $81$.

Let $x\%$ represent the percentage that $81$ is of $9$. The value of $x$ can be found using the proportion $\frac{81}{9}=\frac{x}{100}$ or $9=\frac{x}{100}$.

Multiplying both sides of this equation by $100$ yields $900=x$. Thus, when the number of coins in the collection increased from $9$ to $90$, the percent increase was $900\%$.