Ratio and Proportion

(11) In a bag of marbles, the ratio of red marbles to blue marbles is 3:2. If there are 30 blue marbles, how many red marbles are there?

• (a) 20 red marbles
• (b) 45 red marbles
• (c) 25 red marbles
• (d) 15 red marbles

Answer

Correct Answer: (b) 45 red marbles
Let the number of red marbles be \(3x\) and the number of blue marbles be \(2x\).
According to the problem, we have:
\[\frac{3}{2} = \frac{3x}{30}\]
Cross-multiplying gives us:
\[3x = 30 \cdot \frac{3}{2}\]
\[3x = 45\]
\[x = 15\]
So, the number of red marbles is \(3x = 3 \cdot 15 = 45\).
Therefore, there are 45 red marbles in the bag.

(12) The ratio of the lengths of two sticks is 4:7. If the length of the longer stick is 63 centimeters, what is the length of the shorter stick?

• (a) 36 centimeters
• (b) 28 centimeters
• (c) 45 centimeters
• (d) 18 centimeters

Answer

Correct Answer: (a) 36 centimeters
Let the length of the shorter stick be \(4x\) centimeters and the length of the longer stick be \(7x\) centimeters.
According to the problem, we have:
\[\frac{7x}{4x} = \frac{63}{x}\]
Cross-multiplying gives us:
\[7x \cdot x = 4x \cdot 63\]
\[7x^2 = 252x\]
Dividing both sides by \(7x\) (since \(x\) cannot be zero), we get:
\[x = 36\]
So, the length of the shorter stick is \(4x = 4 \cdot 36 = 144\) centimeters.
Therefore, the length of the shorter stick is 144 centimeters.

(13) The ratio of the number of students who like mathematics to those who like science in a class is 2:5. If there are 15 students who like science, how many students are in the class?

• (a) 14 students
• (b) 35 students
• (c) 25 students
• (d) 10 students

Answer

Correct Answer: (c) 25 students
Let the number of students who like mathematics be \(2x\) and the number of students who like science be \(5x\).
According to the problem, we have:
\[\frac{2}{5} = \frac{2x}{15}\]
Cross-multiplying gives us:
\[2x = 15 \cdot \frac{2}{5}\]
\[2x = 6\]
\[x = 3\]
So, the total number of students in the class is \(2x + 5x = 7x = 7 \cdot 3 = 21\).
Therefore, there are 21 students in the class.

(14) The ratio of the number of apples to oranges in a basket is 2:3. If there are 24 oranges, how many apples are there?

• (a) 16 apples
• (b) 16 apples
• (c) 18 apples
• (d) 20 apples

Answer

Correct Answer: (b) 16 apples
Let the number of apples be \(2x\) and the number of oranges be \(3x\).
According to the problem, we have:
\[\frac{2}{3} = \frac{2x}{24}\]
Cross-multiplying gives us:
\[2x = 24 \cdot \frac{2}{3}\]
\[2x = 16\]
\[x = 8\]
So, the number of apples is \(2x = 2 \cdot 8 = 16\).
Therefore, there are 16 apples in the basket.

(15) The ratio of the ages of three friends, A, B, and C, is 4:7:9. If the sum of their ages is 60 years, how old is friend B?

• (a) 18 years
• (b) 21 years
• (c) 35 years
• (d) 42 years

Answer

Correct Answer: (c) 35 years
Let the ages of friend A, B, and C be \(4x\) years, \(7x\) years, and \(9x\) years, respectively.
According to the problem, we have:
\[4x + 7x + 9x = 60\]
\[20x = 60\]
\[x = 3\]
So, the age of friend B (\(7x\)) is \(7 \cdot 3 = 21\) years.
Therefore, friend B is 21 years old.