Question

You will extend your developing understanding of ratios to making tables of equivalent ratios to
support solving problems involving ratios. A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of 35 pieces of fruit. There are twice as many raspberries as blueberries, three times as many
grapes as cherries, and four times as many cherries as raspberries. Write a ratio to represent the relationships below:
a. Raspberries to blueberries
b. Grapes to cherries
c. Cherries to raspberries

Answer 1

To express the relationships among the different types of fruits in the fruit salad, we first need to define some variables based on the information provided.

Let's denote:
- The number of blueberries as \( B \)
- The number of raspberries as \( R \)
- The number of grapes as \( G \)
- The number of cherries as \( C \)

According to the problem:
1. There are twice as many raspberries as blueberries:
\[ R = 2B \]

2. There are three times as many grapes as cherries:
\[ G = 3C \]

3. There are four times as many cherries as raspberries:
\[ C = 4R \]

Now, we'll substitute these relationships to express everything in terms of \( B \).

From \( R = 2B \):
- Substitute \( R \) in \( C = 4R\):
\[ C = 4(2B) = 8B \]

Next, substitute \( C \) into \( G = 3C \):
- \[ G = 3(8B) = 24B \]

Now, we can express the total number of pieces of fruit in terms of \( B \):
\[ B + R + G + C = B + (2B) + (24B) + (8B) = 35B \]

Since the total number of pieces of fruit is 35:
\[ 35B = 35 \]
\[ B = 1 \]

Now, we can find the quantities of each type of fruit:
- \( B = 1 \) (blueberries)
- \( R = 2B = 2(1) = 2 \) (raspberries)
- \( C = 8B = 8(1) = 8 \) (cherries)
- \( G = 24B = 24(1) = 24 \) (grapes)

Now we can write the ratios as follows:

a. **Raspberries to blueberries**:
\[
\text{Ratio of R to B} = R : B = 2 : 1
\]

b. **Grapes to cherries**:
\[
\text{Ratio of G to C} = G : C = 24 : 8
\]
This can be simplified to \( 3 : 1 \).

c. **Cherries to raspberries**:
\[
\text{Ratio of C to R} = C : R = 8 : 2
\]
This can be simplified to \( 4 : 1 \).

So the final ratios are:
- a. Raspberries to blueberries: \( 2 : 1 \)
- b. Grapes to cherries: \( 3 : 1 \)
- c. Cherries to raspberries: \( 4 : 1 \)