Solve Percent Change - Praxis Core Math

1. The number of defective parts produced by a machine decreased from 250 per day to 200 per day. What is the percentage decrease in the number of defective parts?
   Answer: 20% The formula for percent decrease is [(Original Value - New Value) / Original Value] \times 100%. The decrease is $250 - 200 = 50$. The percentage decrease is $(50 / 250) \times 100\% = 0.20 \times 100\% = 20\%$. A common mistake is to divide by the new value (50/200 = 25%).
2. A real estate agent's commission is 6% of a home's selling price. If the agent's commission was $21,600 for a sale, but this was after the agent gave 10% of their commission back to the buyer, what was the selling price of the home?
   Answer: $400,000 Let C be the agent's full commission. The agent received $21,600, which is 90% of the full commission (100% - 10%). So, $C \times 0.90 = $21,600$. The full commission was $C = $21,600 / 0.90 = $24,000$. This full commission is 6% of the home's selling price (S). So, $S \times 0.06 = $24,000$. The selling price was $S = $24,000 / 0.06 = $400,000$.
3. A stock's value decreased by 30%. What percentage increase is required for the stock to return to its original value before the decrease?
   Answer: 42.9% Let the original value be V. After a 30% decrease, the new value is $V \times (1 - 0.30) = 0.7V$. To return to V, the new value must be multiplied by some factor x: $(0.7V) \times x = V$. Dividing by V gives $0.7x = 1$, so $x = 1 / 0.7 \approx 1.4286$. This multiplier corresponds to an increase of approximately 42.86%, which rounds to 42.9%.
4. A company had 800 employees in January. In February, the number of employees increased by 10%. In March, the number of employees decreased by 10%. How many employees did the company have at the end of March?
   Answer: 792 Number of employees in February: $800 \times (1 + 0.10) = 800 \times 1.10 = 880$. The 10% decrease in March is based on the February number: $880 \times (1 - 0.10) = 880 \times 0.90 = 792$. There were 792 employees at the end of March.
5. A utility bill includes a fixed fee of $20 plus a charge for electricity usage. Last month, the usage charge was $80. This month, the utility company increased the per-unit electricity rate by 15%. If the customer's usage remains the same, what is the percent increase in their total utility bill?
   Answer: 12% Last month's total bill was $20 (fixed) + $80 (usage) = $100$. The 15% rate increase applies only to the usage charge. The new usage charge is $80 \times (1 + 0.15) = $80 \times 1.15 = $92$. The new total bill is $20 (fixed) + $92 (usage) = $112$. The increase in the total bill is $112 - $100 = $12$. The percent increase for the total bill is ($12 / $100) \times 100\% = 12\%.
6. The length of a rectangle is increased by 10% and its width is decreased by 10%. By what percentage is the area of the rectangle changed?
   Answer: It decreases by 1%. Let the original length be L and width be W. Original Area = L \times W. The new length is $1.10 \times L$ and the new width is $0.90 \times W$. The new area is $(1.10 \times L) \times (0.90 \times W) = 0.99 \times L \times W$. The new area is 99% of the original area, which represents a 1% decrease.
7. The value of a house increased by 5% last year. This year, its value increased by 8%. If the house was originally worth $300,000 before the first increase, what is its value now?
   Answer: $340,200 Value after the first year (5% increase): $300,000 \times 1.05 = $315,000$. Value after the second year (8% increase on the new value): $315,000 \times 1.08 = $340,200$. Adding the percentages (13%) and applying it once would give an incorrect answer of $339,000.
8. The price of a product is increased by 10%. Then, the new price is increased by another 10%. The final price is what percentage of the original price?
   Answer: 121% Let the original price be P. The first increase results in a price of $1.10 \times P$. The second increase is applied to this new price: $(1.10 \times P) \times 1.10 = 1.21 \times P$. The final price is 1.21 times the original price, which is 121% of the original price.
9. The base of a triangle is increased by 30% and its height is decreased by 20%. What is the net percent change in the area of the triangle?
   Answer: A 4% increase The area of a triangle is $(1/2) \times base \times height$. Let the original base be B and height be H. The new base is $1.3B$ and the new height is $0.8H$. The new area is $(1/2) \times (1.3B) \times (0.8H) = (1/2) \times (1.04 \times BH) = 1.04 \times Area_{original}$. A multiplier of 1.04 represents a 4% increase.
10. In a school district, 60% of the teachers are female. Of the female teachers, 45% have a master's degree. What percentage of all teachers in the district are females with a master's degree?
    Answer: 27% This is a percent-of-a-percent problem. To find the percentage of the total, multiply the two percentages (as decimals): $0.60 \times 0.45 = 0.27$. Converting this decimal back to a percentage gives 27%.
11. An investor allocates 40% of a portfolio to bonds. Of that bond allocation, 25% is in municipal bonds. The municipal bonds then decrease in value by 5%. This decrease contributes to what percentage decrease for the entire portfolio?
    Answer: 0.5% First, find the percentage of the entire portfolio that is in municipal bonds: $40\% \times 25\% = 0.40 \times 0.25 = 0.10$, or 10%. Next, calculate the impact of the 5% decrease on this portion relative to the whole portfolio: $10\% \times 5\% = 0.10 \times 0.05 = 0.005$. As a percentage, this is 0.5%.
12. Due to increased efficiency, a company increased its output by 25%. To return to its original output level, by what percentage must the new, higher output be decreased?
    Answer: 20% Let the original output be O. The new output is $1.25 \times O$. We need to find the percentage decrease, p, that returns the new output to the original: $(1.25 \times O) \times (1 - p) = O$. Divide both sides by O: $1.25 \times (1 - p) = 1$. Then, $1 - p = 1 / 1.25 = 0.80$. Solving for p gives $p = 1 - 0.80 = 0.20$, which is 20%.
13. If the price of gasoline increases by 25%, by what approximate percentage must a driver reduce their gasoline consumption to keep their total gasoline spending the same?
    Answer: 20% Let P be the original price and C be the original consumption. Spending is $P \times C$. The new price is $1.25P$. Let the new consumption be $C_{new}$. We want the spending to be the same: $(1.25P) \times C_{new} = P \times C$. Dividing by P gives $1.25 \times C_{new} = C$. So, $C_{new} = C / 1.25 = 0.8C$. The new consumption is 80% of the original, which means consumption must be reduced by 20%.
14. A department store offers a 20% discount on all purchases. A customer also has a coupon for an additional 10% off the discounted price. If these discounts are applied sequentially to a purchase of $150, what is the total percent discount the customer receives?
    Answer: 28% First discount: $150 \times (1 - 0.20) = $120$. Second discount on the new price: $120 \times (1 - 0.10) = $108$. The total discount amount is $150 - $108 = $42$. To find the total percent discount, divide the discount amount by the original price: ($42 / $150) \times 100\% = 28\%. A common error is to simply add the percentages (20% + 10% = 30%).
15. The population of a city grew by 5% from 2015 to 2020. If the population was 441,000 in 2020, what was the population in 2015?
    Answer: 420,000 Let P be the 2015 population. The 2020 population is given by $P \times (1 + 0.05) = 441,000$. So, $P \times 1.05 = 441,000$. To find P, we calculate $P = 441,000 / 1.05 = 420,000$. A common mistake is to calculate 5% of 441,000 and subtract it, which would give an incorrect answer.
16. The number of employees at a company decreased by 20% in one year. The following year, the number of employees increased by 20%. If there were 100 employees initially, how many employees were there after the two changes?
    Answer: 96 Initially, there are 100 employees. A 20% decrease leaves $100 \times (1-0.20) = 80$ employees. The following year, a 20% increase is calculated on the new number of employees: $80 \times (1+0.20) = 80 \times 1.2 = 96$ employees. The common mistake is to assume the changes cancel each other out.
17. A student's score on a test improved from 60 to 75. What was the percentage increase in the student's score?
    Answer: 25% The formula for percent increase is [(New Value - Original Value) / Original Value] \times 100%. The increase in score is $75 - 60 = 15$. The percent increase is $(15 / 60) \times 100\% = 0.25 \times 100\% = 25\%$. A common error is to divide by the new value (15/75 = 20%).
18. A television is on sale for $357 after a 15% discount. What was the original price of the television?
    Answer: $420.00 Let P be the original price. The sale price is the original price minus the 15% discount, so $P \times (1 - 0.15) = $357$. This simplifies to $P \times 0.85 = $357$. To find the original price, divide the sale price by 0.85: $P = $357 / 0.85 = $420$.
19. A scientist measures the length of a specimen to be 8.1 cm. The actual length is 7.5 cm. What is the percent error in the measurement, rounded to the nearest tenth of a percent?
    Answer: 8.0% Percent error is calculated as $|(\text{Measured Value} - \text{Actual Value}) / \text{Actual Value}| \times 100\%$. The difference is $|8.1 - 7.5| = 0.6$ cm. The percent error is $(0.6 / 7.5) \times 100\% = 0.08 \times 100\% = 8.0\%$. A common error is to divide by the measured value (8.1 cm), which gives an incorrect result.
20. A restaurant bill is $120. A 15% tip is added, and then an 8% tax is calculated on the original bill amount. What is the total amount paid?
    Answer: $147.60 This problem requires calculating the tip and tax separately and adding them to the original bill. Tip: $120 \times 0.15 = $18$. Tax: $120 \times 0.08 = $9.60$. Total amount: $120 + $18 + $9.60 = $147.60$. Note that if the tax were calculated on the bill plus tip, the answer would be different ($138 \times 1.08 = $149.04), which is a distractor.

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