The mean of five math tests is 84 . The median is 86 . Two of the test grades are 70 and one test grade is 100 . What are the other two grades?

1 answer

To find the two unknown test grades, we can denote them as \( x \) and \( y \).

We know the following:

  1. The mean of the five tests is 84, which means: \[ \frac{70 + 70 + 100 + x + y}{5} = 84 \] Multiplying both sides by 5: \[ 70 + 70 + 100 + x + y = 420 \] Simplifying the left side: \[ 240 + x + y = 420 \] Thus: \[ x + y = 180 \quad \text{(Equation 1)} \]

  2. The median of the five test scores is 86. To maintain the median of 86, when the grades are arranged in increasing order, the middle value (third in a sorted list) should be 86. So, our arrangement should include 70, 70, and 100, along with \( x \) and \( y \).

Considering \( x \) and \( y \), \( x \) and \( y \) must be between 70 and 100 to keep the median at 86. Therefore, when we list the grades in ascending order, they must fit within this structure.

Assuming \( x \leq y \), the possible arrangements could be:

  • If both \( x \) and \( y \) are less than 86: the arrangement would be \( 70, 70, x, y, 100 \)
  • If one of \( x \) or \( y \) is 86: the arrangement could be \( 70, 70, 86, y, 100 \)
  • If both grades are greater than or equal to 86: the arrangement would need careful analysis since the median is the third number.

Since we know the median is 86:

  • We explore the case where \( x = 86 \) and \( y \) is undetermined, or vice versa.

Substituting \( x = 86 \) into Equation 1: \[ 86 + y = 180 \] Thus: \[ y = 180 - 86 = 94 \]

The possible grades are \( 70, 70, 86, 94, 100 \).

To check:

  • The mean: \[ \frac{70 + 70 + 86 + 94 + 100}{5} = \frac{420}{5} = 84 \]
  • The median is the third value:
    • Ordered: \( 70, 70, 86, 94, 100 \), the third value is indeed 86.

Therefore, the other two grades are \( \boxed{86} \) and \( \boxed{94} \).