Percent Change and Growth - Quantitative Reasoning

1. If a value decreases by $p\%$ each period for $n$ periods, state the decay formula from $A$.

   $A\left(1-\frac{p}{100}\right)^n$. Applies the decay multiplier compounded over $n$ periods to the initial value $A$.

2. State the relationship between percent change and the ratio $\frac{B}{A}$.

   $\text{percent change}=\left(\frac{B}{A}-1\right)\times 100\%$. Subtracts $1$ from the ratio $\frac{B}{A}$ and multiplies by $100$ to get percent change.

3. A value is halved. What is the percent decrease?

   $50\%$. Halving corresponds to losing half of the original value, which is a $50\%$ decrease.

4. What is the percent change from $40$ to $50$?

   $25\%$. The difference of $10$ divided by the original $40$ yields $0.25$, or $25\%$.

5. Find the overall multiplier for successive increases of $+20\%$ and $+30\%$.

   $1.56$. Multiplies the individual multipliers $1.2$ and $1.3$ for the successive increases.

6. A value increases by $25\%$; by what percent must it then decrease to return to the original value?

   $20\%$. To reverse a $25\%$ increase, the subsequent decrease is relative to the higher value, requiring $20\%$.

7. A value decreases by $20\%$; by what percent must it then increase to return to the original value?

   $25\%$. To reverse a $20\%$ decrease, the subsequent increase is relative to the lower value, requiring $25\%$.

8. If a value is multiplied by $k$, what is the percent change in terms of $k$?

   $(k-1)\times 100\%$. Subtracts $1$ from $k$ to find the relative change, then multiplies by $100$ for percent.

9. A store marks an item $20\%$ off, then raises the discounted price by $20\%$. What is the net percent change?

   $-4\%$. The overall multiplier of $0.8 \times 1.2 = 0.96$ represents a $4\%$ decrease.

10. State the multiplier for a decrease of $p\%$ (in terms of $p$).

    $1-\frac{p}{100}$. Subtracts the fractional percent decrease from 1 to form the scaling factor.

11. A value doubles. What is the percent increase?

    $100\%$. Doubling corresponds to a multiplier of $2$, which is a $100\%$ increase from the original.

12. State the formula for percent change from an original value $A$ to a new value $B$.

    $\frac{B-A}{A}\times 100\%$. Calculates the relative difference as a fraction of the original value, then converts to percent.

13. State the percent change from $A$ to $B$ when $A=0$.

    Undefined (division by $0$). The percent change formula involves division by $A$, which is zero, making it undefined.

14. If a value increases by $p\%$ each period for $n$ periods, state the growth formula from $A$.

    $A\left(1+\frac{p}{100}\right)^n$. Applies the growth multiplier compounded over $n$ periods to the initial value $A$.

15. If a price increases by $15\%$, what is the multiplier applied to the original price?

    $1.15$. Adds $15\%$ as a decimal to $1$ to scale the original price.

16. Find the overall multiplier for successive changes of $+10\%$ then $-10\%$.

    $0.99$. Multiplies the individual multipliers $1.1$ and $0.9$ for the successive changes.

17. What is the percent change from $50$ to $40$?

    $-20\%$. The difference of $-10$ divided by the original $50$ yields $-0.2$, or $-20\%$.

18. What is the net percent change for successive changes of $+10\%$ then $-10\%$?

    $-1\%$. The overall multiplier of $0.99$ represents a $1\%$ decrease from the original.

19. What is the percent decrease from $50$ to $40$?

    $20\%$. The absolute difference of $10$ divided by the original $50$ yields $0.2$, or $20\%$ decrease.

20. An investment of $200$ decreases $10\%$ each year for $2$ years. What is the final value?

    $162$. Compounds the $10\%$ decrease by multiplying $200$ by $0.9^2$.

21. What is the net percent change for successive increases of $+20\%$ and $+30\%$?

    $56\%$. The overall multiplier of $1.56$ indicates a $56\%$ increase from the original.

22. If a quantity decreases by $30\%$, what is the multiplier applied to the original quantity?

    $0.70$. Subtracts $30\%$ as a decimal from $1$ to scale the original quantity.

23. What is the percent increase from $80$ to $100$?

    $25\%$. The difference of $20$ divided by the original $80$ yields $0.25$, or $25\%$.

24. State the multiplier for an increase of $p\%$ (in terms of $p$).

    $1+\frac{p}{100}$. Adds the fractional percent increase to 1 to form the scaling factor.

25. A population is $1000$ and grows $5\%$ for $2$ years. What is the final population?

    $1102.5$. Compounds the $5\%$ growth by multiplying $1000$ by $1.05^2$.

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