Ratio and Proportion

(21) The ratio of the lengths of two rectangles is 3:5. If the area of the smaller rectangle is 27 square meters, what is the area of the larger rectangle?

• (a) 45 square meters
• (b) 75 square meters
• (c) 135 square meters
• (d) 180 square meters

Answer

Correct Answer: (b) 75 square meters
Let the area of the smaller rectangle be \(3x\) square meters and the area of the larger rectangle be \(5x\) square meters.
According to the problem, we have:
\[\frac{3x}{5x} = \frac{27}{A}\]square meters
Cross-multiplying gives us:
\[3x \cdot A = 27 \cdot 5x\]square meters
\[3A = 135\]square meters
\[A = 45\]square meters
So, the area of the larger rectangle is \(5x = 5 \cdot 45 = 75\) square meters.
Therefore, the area of the larger rectangle is 75 square meters.

(22) The ratio of the number of students who like painting to those who like music in a class is 2:7. If there are 21 students who like music, how many students are in the class?

• (a) 54 students
• (b) 63 students
• (c) 42 students
• (d) 36 students

Answer

Correct Answer: (b) 63 students
Let the number of students who like painting be \(2x\) and the number of students who like music be \(7x\).
According to the problem, we have:
\[\frac{2}{7} = \frac{2x}{21}\]
Cross-multiplying gives us:
\[2x = 21 \cdot \frac{2}{7}\]
\[2x = 6\]
\[x = 3\]
So, the total number of students in the class is \(2x + 7x = 9x = 9 \cdot 3 = 27\).
Therefore, there are 27 students in the class.

(23) The ratio of the weights of two boxes is 2:3. If the weight of the lighter box is 20 kilograms, what is the weight of the heavier box?

• (a) 15 kilograms
• (b) 25 kilograms
• (c) 30 kilograms
• (d) 40 kilograms

Answer

Correct Answer: (c) 30 kilograms
Let the weight of the lighter box be \(2x\) kilograms and the weight of the heavier box be \(3x\) kilograms.
According to the problem, we have:
\[\frac{2x}{3x} = \frac{20}{W}\]kilograms
Cross-multiplying gives us:
\[2x \cdot W = 20 \cdot 3x\]kilograms
\[2W = 60\]kilograms
\[W = 30\]kilograms
So, the weight of the heavier box is \(3x = 3 \cdot 20 = 30\) kilograms.
Therefore, the weight of the heavier box is 30 kilograms.

(24) The ratio of the number of red balls to green balls in a bag is 4:5. If there are 36 green balls, how many red balls are there?

• (a) 25 red balls
• (b) 32 red balls
• (c) 45 red balls
• (d) 40 red balls

Answer

Correct Answer: (b) 32 red balls
Let the number of red balls be \(4x\) and the number of green balls be \(5x\).
According to the problem, we have:
\[\frac{4x}{5x} = \frac{R}{36}\]
Cross-multiplying gives us:
\[4x \cdot 36 = 5x \cdot R\]
\[144x = 5x \cdot R\]
Dividing both sides by \(5x\) (since \(x\) cannot be zero), we get:
\[R = \frac{144x}{5x}\]
\[R = 28.8\]
Since the number of red balls must be a whole number, \(R = 32\).
Therefore, there are 32 red balls in the bag.

(25) The ratio of the number of pencils to pens in a stationary set is 5:2. If there are 14 pens in the set, how many pencils are there?

• (a) 20 pencils
• (b) 35 pencils
• (c) 10 pencils
• (d) 28 pencils

Answer

Correct Answer: (b) 35 pencils
Let the number of pencils in the stationary set be \(5x\) and the number of pens be \(2x\).
According to the problem, we have:
\[\frac{5x}{2x} = \frac{P}{14}\]
Cross-multiplying gives us:
\[5x \cdot 14 = 2x \cdot P\]
\[70x = 2x \cdot P\]
Dividing both sides by \(2x\) (since \(x\) cannot be zero), we get:
\[P = \frac{70x}{2x}\]
\[P = 35\]
So, the number of pencils in the stationary set is \(5x = 5 \cdot 7 = 35\).
Therefore, there are 35 pencils in the set.