Several word problems on percentages with clear explanations and detailed solutions are presented below.
The original price of a shirt was \(\$20\). It was decreased to \(\$15\). What is the percent decrease of the price of this shirt?
Solution
The absolute decrease is
\[ 20 - 15 = 5 \]The percent decrease is the absolute decrease divided by the original price (part/whole):
\[ \frac{5}{20} = 0.25 \]Converting to percent:
\[ 0.25 \times 100\% = 25\% \]Mary has a monthly salary of \(\$1200\). She spends \(\$280\) per month on food. What percent of her monthly salary does she spend on food?
Solution
The fraction of salary spent on food is
\[ \frac{280}{1200} = 0.2333 \approx 0.23 \]Converting to percent:
\[ 0.23 \times 100\% = 23\% \]The price of a pair of trousers was decreased by 22% to \(\$30\). What was the original price of the trousers?
Solution
Let \(x\) be the original price and \(y\) the absolute decrease. Then
\[ x - y = 30 \]Since \(y = 22\%\) of \(x\):
\[ y = 0.22x \]Substitute into the first equation:
\[ x - 0.22x = 30 \] \[ 0.78x = 30 \quad \Rightarrow \quad x = 38.46 \]The original price was approximately \(\$38.50\).
The price of an item changed from \(\$120\) to \(\$100\). Then later the price decreased again from \(\$100\) to \(\$80\). Which of the two decreases was larger in percentage terms?
Solution
First decrease in percent:
\[ \frac{120 - 100}{120} = \frac{20}{120} \approx 16.7\% \]Second decrease in percent:
\[ \frac{100 - 80}{100} = \frac{20}{100} = 20\% \]The second decrease is larger in percent terms because the whole (original price) is smaller.
The price of an item decreased by 20% to \(\$200\). Then later the price decreased again from \(\$200\) to \(\$150\). What is the percent decrease from the original price to the final price?
Solution
Let \(x\) be the original price:
\[ x - 0.20x = 200 \quad \Rightarrow \quad 0.8x = 200 \quad \Rightarrow \quad x = 250 \]The total percent decrease is:
\[ \frac{250 - 150}{250} = \frac{100}{250} = 0.4 = 40\% \]A number increases from 30 to 40 and then decreases from 40 to 30. Compare the percent increase and percent decrease.
Solution
Percent increase:
\[ \frac{40 - 30}{30} = \frac{10}{30} = 0.333 \approx 33\% \]Percent decrease:
\[ \frac{40 - 30}{40} = \frac{10}{40} = 0.25 = 25\% \]In absolute terms, the percent decrease is less than the percent increase because the base values differ.
A family had dinner in a restaurant and paid \(\$30\) for food. They also had to pay 9.5% sales tax and 10% tip. How much did they pay in total?
Solution
\[ 30 + 0.095 \times 30 + 0.10 \times 30 = 30 + 2.85 + 3 = \$35.85 \]Total paid: \(\$35.85\)
A shop offers shirts at \(\$20\) each. Buying 2 shirts gives a 15% discount on both shirts and an extra 10% discount on the second shirt. How much would one pay for two shirts?
Solution
Price of each of the shirts after the first discount :
\[ 20 - 0.15 \times 20 = 17 \]Price of second shirt after 10% discount:
\[ 17 - 0.10 \times 17 = 15.3 \]Total cost for two shirts:
\[ 17 + 15.3 = 32.3 \]Smith invested \(\$5000\) for two years. For the first year, the rate of interest was 7% and the second year it was 8.5%. How much interest did he earn at the end of the two year period?
Solution
Interest for first year:
\[ 0.07 \times 5000 = 350 \]Interest for second year (on principal + first year interest):
\[ 0.085 \times (5000 + 350) = 454.75 \]Total interest:
\[ 350 + 454.75 = 804.75 \]Janette invested \(\$2000\) at 5% compounded annually for 5 years. How much interest did she earn at the end of 5 years?
Solution
Principal at the end of 5 years:
\[ P_5 = 2000 \times (1 + 0.05)^5 = 2000 \times 1.27628 \approx 2552.56 \]Interest earned:
\[ 2552.56 - 2000 = 552.56 \]Tom borrowed \(\$600\) at 10% simple interest for 3 years. How much does he repay at the end of 3 years?
Solution
\[ \text{Interest} = 600 \times 0.10 \times 3 = 180 \] \[ \text{Total repayment} = 600 + 180 = 780 \]World population: 6.6 billion, with 1.2 billion in richer countries growing at 0.25% per year and 5.4 billion in less developed countries growing at 1.5% per year. What will the total population be in 5 years? (round to 3 significant digits)
Solution
\[ P_R = 1.2 \times (1 + 0.0025)^5 \] \[ P_L = 5.4 \times (1 + 0.015)^5 \] \[ P_\text{world} = P_R + P_L \approx 7.03 \text{ billion} \]Cassandra invested \(\$10,000\) in two parts: one at 7.5%, the other at 8.5%. Income from both investments is \(\$820\). How much was invested at each rate?
Solution
Let \( x \) and \( y \) be the amounts invested at 7.5% and 8.5% respectively. \[ \begin{cases} x + y = 10000 \\ 0.075x + 0.085y = 820 \end{cases} \] Solve system of equations. \[ x = 3000, \quad y = 7000 \]The monthly salary \(S\) of a shop assistant is a fixed \(\$500\) plus 5% of all monthly sales. What should monthly sales be so that her salary reaches \(\$1500\)?
Solution
Let \( x \) be the monthly sales. \[ S = 500 + 0.05x \] \[ 1500 = 500 + 0.05x \quad \Rightarrow \quad x = 20000 \]A chemist has 20% and 40% acid solutions. What amounts of each should be used to make 300 ml of 28% acid solution?
Solution
Let \( x \) and \( y \) be the solutions at 20% and 40% respectively. \[ x + y = 300 \] \[ 0.20x + 0.40y = 0.28 \times 300 \] Solve system of equations. \[ x = 180, \quad y = 120 \]What percent of the total area of the circular disk is colored red?
Solution
Total area of disk \[ A_\text{disk} = \pi r^2 \] Angle \( t \) in radians of central angle of red sector \[ t = (360 - 120) \frac{\pi}{180} = \frac{4}{3}\pi \] Area of red part \[ A_\text{red} = \frac{1}{2} t r^2 \] \[ P = \frac{A_\text{red}}{A_\text{disk}} = \frac{\frac{1}{2} \cdot \frac{4}{3}\pi r^2}{\pi r^2} = \frac{4}{6} \approx 66.7\% \]What percent of the total area of the rectangle is colored red?
Solution
Area of rectangle \[ A_\text{rect} = L \times W \] Area of red triangle \[ A_\text{triangle} = \frac{1}{2} \times L \times \frac{W}{2} = \frac{1}{4} LW \] Percentage of area of red triangle (out of the total area) \[ P = \frac{A_\text{triangle}}{A_\text{rect}} = \frac{1}{4} = 25\% \]