Set 3: Ratio and Proportion (Harder)
1.
A’s income is 20% less than B’s. B’s income is 25% less than C’s. What percent of C’s income is A’s income?
Solution (B):
Let C = 100.
B is 25% less than C = 75.
A is 20% less than B = \(0.8 \times 75 = 60\).
A is 60, C is 100. So A is 60% of C.
2.
A jar contains red and blue marbles in a ratio of 3:5. If 20 blue marbles are added, the ratio becomes 3:7. How many red marbles are in the jar?
Solution (D):
Let Red = \(3x\), Blue = \(5x\).
New Blue = \(5x + 20\).
New Ratio: \( \frac{3x}{5x+20} = \frac{3}{7} \).
\( 21x = 3(5x + 20) \implies 21x = 15x + 60 \implies 6x = 60 \implies x = 10 \).
Red marbles = \(3(10) = 30\).
3.
If a city’s population grows by 10% per year, and the current population is 20,000, what will be the population in 2 years?
Solution (C):
Year 1: \( 20,000 \times 1.1 = 22,000 \).
Year 2: \( 22,000 \times 1.1 = 24,200 \).
4.
Fresh mushrooms consist of 90% water, while dried mushrooms consist of 10% water. How many kilograms of fresh mushrooms are needed to make 2 kg of dried mushrooms?
Solution (A):
Solid content in dried: \(90\% \text{ of } 2\text{kg} = 1.8\text{kg}\).
In fresh, solid content is 10%.
\(0.10 \times \text{Fresh} = 1.8\).
\(\text{Fresh} = \frac{1.8}{0.1} = 18\text{kg}\).
5.
Jane scored 80 on her first test. Her second score was 10% higher than the first. Her third score was 10% lower than the second. What was her third score?
Solution (E):
Test 1: 80.
Test 2: \(80 \times 1.1 = 88\).
Test 3: \(88 \times 0.9 = 79.2\).
6.
The ratio of boys to girls in a club is 3:4. If 4 boys leave and 4 girls join, the ratio becomes 1:2. How many members were there originally?
Solution (B):
Original: \(3x\) boys, \(4x\) girls. Total = \(7x\).
New: \(3x-4\) boys, \(4x+4\) girls.
Ratio: \( \frac{3x-4}{4x+4} = \frac{1}{2} \).
\( 2(3x-4) = 1(4x+4) \implies 6x – 8 = 4x + 4 \implies 2x = 12 \implies x = 6 \).
Original total = \(7x = 7(6) = 42\).
7.
A person spends 30% of their income on rent and 40% of the **remainder** on food. What percentage of the original income is spent on food?
Solution (D):
Let income = 100.
Rent = 30. Remainder = 70.
Food = 40% of 70 = \(0.4 \times 70 = 28\).
Food is 28% of the original income.
8.
An alloy contains copper and zinc in the ratio 5:3. If there are 24 kg of zinc, how much copper is in the alloy?
Solution (C):
Ratio \( \frac{C}{Z} = \frac{5}{3} \).
\( \frac{C}{24} = \frac{5}{3} \implies C = \frac{5}{3} \times 24 = 5 \times 8 = 40 \).
9.
If the price of a book increases from $25 to $30, what is the percent increase?
Solution (A):
Increase = \(30 – 25 = 5\).
Percent = \( \frac{5}{25} \times 100\% = \frac{1}{5} \times 100\% = 20\% \).
10.
On a map, 1 inch represents 50 miles. If two cities are 3.5 inches apart on the map, how far apart are they in reality?
Solution (B):
\( 1 \text{ in} : 50 \text{ mi} \).
\( 3.5 \text{ in} \times 50 \text{ mi/in} = 175 \text{ miles} \).
Score: 0 / 0