Ratio and Proportion

(41) The ratio of the number of red balls to blue balls in a bag is 3:2. If there are 20 blue balls in the bag, how many red balls are there?

• (a) 12 red balls
• (b) 30 red balls
• (c) 15 red balls
• (d) 40 red balls

Answer

Correct Answer: (b) 30 red balls
Let the number of red balls in the bag be \(3x\) and the number of blue balls be \(2x\).
According to the problem, we have:
\[\frac{3x}{2x} = \frac{R}{20}\]
Cross-multiplying gives us:
\[3x \cdot 20 = 2x \cdot R\]
\[60x = 2x \cdot R\]
Dividing both sides by \(2x\) (since \(x\) cannot be zero), we get:
\[R = \frac{60x}{2x}\]
\[R = 30\]
So, the number of red balls in the bag is \(3x = 3 \cdot 10 = 30\).
Therefore, there are 30 red balls in the bag.

(42) The ratio of the number of marbles to pebbles in a jar is 6:5. If there are 25 pebbles in the jar, how many marbles are there?

• (a) 20 marbles
• (b) 30 marbles
• (c) 15 marbles
• (d) 35 marbles

Answer

Correct Answer: (b) 30 marbles
Let the number of marbles in the jar be \(6x\) and the number of pebbles be \(5x\).
According to the problem, we have:
\[\frac{6x}{5x} = \frac{M}{25}\]
Cross-multiplying gives us:
\[6x \cdot 25 = 5x \cdot M\]
\[150x = 5x \cdot M\]
Dividing both sides by \(5x\) (since \(x\) cannot be zero), we get:
\[M = \frac{150x}{5x}\]
\[M = 30\]
So, the number of marbles in the jar is \(6x = 6 \cdot 5 = 30\).
Therefore, there are 30 marbles in the jar.

(43) The ratio of the ages of three siblings is 4:7:9. If the youngest sibling is 9 years old, how old is the oldest sibling?

• (a) 12 years old
• (b) 15 years old
• (c) 18 years old
• (d) 21 years old

Answer

Correct Answer: (d) 21 years old
Let the age of the youngest sibling be \(4x\) years, the middle sibling be \(7x\) years, and the oldest sibling be \(9x\) years.
According to the problem, we have:
\[\frac{4x}{7x} = \frac{9}{O}\]years
Cross-multiplying gives us:
\[4x \cdot O = 7x \cdot 9\]years
\[4O = 63\]years
\[O = \frac{63}{4}\]years
\[O = 15.75\]years
Since the age of a person must be a whole number, we round to the nearest whole number: \(O = 16\).
So, the age of the oldest sibling is \(9x = 9 \cdot 1.75 = 15.75\), which we round to 16 years.
Therefore, the oldest sibling is 16 years old.

(44) The ratio of the number of students who like math to the number of students who like science in a class is 5:3. If there are 24 students who like science, how many students like math?

• (a) 30 students
• (b) 40 students
• (c) 40 students
• (d) 20 students

Answer

Correct Answer: (c) 40 students
Let the number of students who like math be \(5x\) and the number of students who like science be \(3x\).
According to the problem, we have:
\[\frac{5x}{3x} = \frac{M}{24}\]
Cross-multiplying gives us:
\[5x \cdot 24 = 3x \cdot M\]
\[120x = 3x \cdot M\]
Dividing both sides by \(3x\) (since \(x\) cannot be zero), we get:
\[M = \frac{120x}{3x}\]
\[M = 40\]
So, the number of students who like math is \(5x = 5 \cdot 8 = 40\).
Therefore, there are 40 students who like math.

(45) The ratio of the side lengths of two similar triangles is 2:5. If the shorter triangle has a side length of 6 cm, what is the length of the corresponding side in the longer triangle?

• (a) 15 cm
• (b) 12 cm
• (c) 20 cm
• (d) 15 cm

Answer

Correct Answer: (d) 15 cm
Let the length of the corresponding side in the longer triangle be \(2x\) cm.
According to the problem, we have:
\[\frac{2x}{6} = \frac{5}{L}\]cm
Cross-multiplying gives us:
\[2x \cdot L = 6 \cdot 5\]cm
\[2xL = 30\]cm
\[L = \frac{30}{2x}\]cm
\[L = 15\]cm
So, the length of the corresponding side in the longer triangle is \(2x = 2 \cdot 7.5 = 15\) cm.
Therefore, the length of the corresponding side in the longer triangle is 15 cm.