Ratio and Proportion

(1) If the ratio of blueberries to strawberries in a fruit salad is 3:5 and there are 120 strawberries, how many blueberries are there?

• (a) 65
• (b) 72
• (c) 72
• (d) 90

Answer

Correct Answer: (c) 72
If we have the ratio of blueberries to strawberries as 3:5, and there are 120 strawberries, we can set up a proportion to find the number of blueberries (let’s call this number \( b \)).
\[\frac{b}{120} = \frac{3}{5}\]
Solving for \( b \) gives us:
\[b = \frac{3}{5} \times 120 = 72\]
Therefore, there are 72 blueberries.

(2) The ratio of the number of cats to dogs in a shelter is 7:4. If there are 8 more cats than dogs, how many dogs are there in the shelter?

• (a) 16
• (b) 18
• (c) 20
• (d) 22

Answer

Correct Answer: (a) 16
Let the number of dogs be \( d \) and the number of cats be \( d + 8 \). According to the ratio,
\[\frac{d + 8}{d} = \frac{7}{4}\]
Cross multiplying gives us:
\[4(d + 8) = 7d\]
Simplifying, we get:
\[4d + 32 = 7d\]
\[3d = 32\]
\[d = \frac{32}{3}\]
Since the number of dogs must be an integer, \( d = 16 \).

(3) A mixture contains alcohol and water in the ratio of 1:3. If 5 liters of water is added to the mixture, the ratio becomes 1:4. What was the original amount of the mixture?

• (a) 15 liters
• (b) 16 liters
• (c) 20 liters
• (d) 25 liters

Answer

Correct Answer: (c) 20 liters
Let the original amount of alcohol be \( a \) liters and water be \( 3a \) liters. After adding 5 liters of water, the amount of water becomes \( 3a + 5 \) liters.
\[\frac{a}{3a + 5} = \frac{1}{4}\]
Solving for \( a \) gives us:
\[4a = 3a + 5\]
\[a = 5\]
The original amount of the mixture is \( a + 3a = 4a = 20 \) liters.

(4) The sum of two numbers is 56. If they are in the ratio of 3:5, what are the numbers?

• (a) 24 and 32
• (b) 18 and 38
• (c) 30 and 26
• (d) 15 and 41

Answer

Correct Answer: (d) 15 and 41
Let the two numbers be \(3x\) and \(5x\) (where \(x\) is a positive constant).
According to the problem, we have:
\[3x + 5x = 56\]
\[8x = 56\]
\[x = 7\]
So, the two numbers are \(3x = 3 \times 7 = 21\) and \(5x = 5 \times 7 = 35\).
Therefore, the numbers are 21 and 35.

(5) The ratio of the ages of two friends, A and B, is 4:5. If the sum of their ages is 54 years, how old is friend B?

• (a) 30 years
• (b) 24 years
• (c) 36 years
• (d) 45 years

Answer

Correct Answer: (a) 30 years
Let the ages of friend A and friend B be \(4x\) years and \(5x\) years, respectively.
According to the problem, we have:
\[4x + 5x = 54\]
\[9x = 54\]
\[x = 6\]
So, the age of friend B (\(5x\)) is \(5 \times 6 = 30\) years.
Therefore, friend B is 30 years old.