Ratio and Proportion

(31) The ratio of the lengths of two pieces of string is 7:3. If the length of the shorter piece is 15 meters, what is the length of the longer piece?

• (a) 21 meters
• (b) 45 meters
• (c) 35 meters
• (d) 30 meters

Answer

Correct Answer: (b) 45 meters
Let the length of the shorter piece of string be \(7x\) meters and the length of the longer piece be \(3x\) meters.
According to the problem, we have:
\[\frac{7x}{3x} = \frac{15}{L}\]meters
Cross-multiplying gives us:
\[7x \cdot L = 3x \cdot 15\]meters
\[7L = 45\]meters
\[L = 45\]meters
So, the length of the longer piece is \(3x = 3 \cdot 15 = 45\) meters.
Therefore, the length of the longer piece is 45 meters.

(32) The ratio of the number of chocolates to candies in a jar is 4:9. If there are 36 candies in the jar, how many chocolates are there?

• (a) 24 chocolates
• (b) 16 chocolates
• (c) 12 chocolates
• (d) 18 chocolates

Answer

Correct Answer: (b) 16 chocolates
Let the number of chocolates in the jar be \(4x\) and the number of candies be \(9x\).
According to the problem, we have:
\[\frac{4x}{9x} = \frac{C}{36}\]
Cross-multiplying gives us:
\[4x \cdot 36 = 9x \cdot C\]
\[144x = 9x \cdot C\]
Dividing both sides by \(9x\) (since \(x\) cannot be zero), we get:
\[C = \frac{144x}{9x}\]
\[C = 16\]
So, the number of chocolates in the jar is \(4x = 4 \cdot 4 = 16\).
Therefore, there are 16 chocolates in the jar.

(33) The ratio of the number of students who like pizza to those who like burgers in a class is 3:2. If there are 20 students who like burgers, how many students are in the class?

• (a) 10 students
• (b) 15 students
• (c) 30 students
• (d) 25 students

Answer

Correct Answer: (c) 30 students
Let the number of students who like pizza be \(3x\) and the number of students who like burgers be \(2x\).
According to the problem, we have:
\[\frac{3x}{2x} = \frac{P}{20}\]
Cross-multiplying gives us:
\[3x \cdot 20 = 2x \cdot P\]
\[60x = 2x \cdot P\]
Dividing both sides by \(2x\) (since \(x\) cannot be zero), we get:
\[P = \frac{60x}{2x}\]
\[P = 30\]
So, the total number of students in the class is \(3x + 2x = 5x = 5 \cdot 6 = 30\).
Therefore, there are 30 students in the class.

(34) The ratio of the weights of two boxes is 5:7. If the weight of the lighter box is 20 kilograms, what is the weight of the heavier box?

• (a) 28 kilograms
• (b) 25 kilograms
• (c) 28 kilograms
• (d) 30 kilograms

Answer

Correct Answer: (c) 28 kilograms
Let the weight of the lighter box be \(5x\) kilograms and the weight of the heavier box be \(7x\) kilograms.
According to the problem, we have:
\[\frac{5x}{7x} = \frac{20}{W}\]kilograms
Cross-multiplying gives us:
\[5x \cdot W = 7x \cdot 20\]kilograms
\[5W = 140\]kilograms
\[W = \frac{140}{5}\]kilograms
\[W = 28\]kilograms
So, the weight of the heavier box is \(7x = 7 \cdot 4 = 28\) kilograms.
Therefore, the weight of the heavier box is 28 kilograms.

(35) The ratio of the incomes of two individuals is 2:5. If the first person's income is $600, what is the second person's income?

• (a) $2400
• (b) $1500
• (c) $1500
• (d) $1200

Answer

Correct Answer: (c) $1500
Let the income of the first person be \(2x\) dollars and the income of the second person be \(5x\) dollars.
According to the problem, we have:
\[\frac{2x}{5x} = \frac{600}{I}\]dollars
Cross-multiplying gives us:
\[2x \cdot I = 5x \cdot 600\]dollars
\[2I = 3000\]dollars
\[I = \frac{3000}{2}\]dollars
\[I = 1500\]dollars
So, the income of the second person is \(5x = 5 \cdot 300 = $1500\).
Therefore, the second person's income is $1500.