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Constructing the two way table, what is the total number of seventh graders who prefer bananas?

7th graders : Apples=41 bananas= blank oranges= blank total horizontal=120
8th graders; Apples= blank Bananas=32 oranges=45
Total vertically Apples: 75 Bananas= blank oranges= 72

1 answer

To find the total number of seventh graders who prefer bananas, we will use the information provided in the two-way table. We know:

  • Total number of seventh graders: 120
  • Seventh graders who prefer apples: 41
  • Total number of apples preferred by both grades: 75

First, let's calculate the number of seventh graders who prefer bananas and oranges:

  1. Find the number of seventh graders who prefer apples, bananas, and oranges.

    • Let \( b \) be the number of seventh graders who prefer bananas.
    • Let \( o \) be the number of seventh graders who prefer oranges.

    From the total for seventh graders:

    \[ 41 + b + o = 120 \]

    This simplifies to:

    \[ b + o = 120 - 41 = 79 \quad \text{(1)} \]

  2. Looking at the total preferences for apples:

    • The total number of apples is 75, so for eighth graders, we can calculate their preference as follows:
      • Let \( x \) be the number of eighth graders who prefer apples.

    Thus,

    \[ 41 + x = 75 \]

    Solving for \( x \):

    \[ x = 75 - 41 = 34 \]

  3. Now, we have the information for eighth graders:

    • Apples: 34
    • Bananas: 32
    • Oranges: 45

    Total for eighth graders:

    \[ 34 + 32 + 45 = 111 \]

We validate how many seventh graders eat bananas and oranges:

  1. Total bananas across both grades is the count of bananas preferred by both grades which we have as 32 from eighth graders. So the total number for bananas is \( b + 32 \).

  2. Looking at the total preferences for bananas:

    \[ b + 32 \quad \text{(total bananas)} \]

  3. The total number of bananas preferred is not provided. However, since seventh graders total their fruit preference to 79, we can revise the known from their total:

    Using equation (1):

    \[ b + o = 79 \]

Now let's calculate using the total oranges:

  • The total oranges were provided across both grades, 72:

    The number of eighth graders' oranges is 45, hence seventh graders:

    \[ o = 72 - 45 = 27 \]

  1. Substitute \( o = 27 \) back into equation (1):

    \[ b + 27 = 79 \]

    Solving for \( b \):

    \[ b = 79 - 27 = 52 \]

Thus, the total number of seventh graders who prefer bananas is \( \boxed{52} \).