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:
-
There are twice as many raspberries as blueberries: \[ R = 2B \]
-
There are three times as many grapes as cherries: \[ G = 3C \]
-
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: \( 4 : 1 \)