AMC 8 · 2016 · #3

Easy mode Grade 6

📗 View original problem →

Problem

Four students take an exam. Three of them scored $70$ , $80$ , and $90$ .

When you average all four scores together, the average comes out to $70$ .

What did the fourth student score?

$\textbf{(A) }40\qquad\textbf{(B) }50\qquad\textbf{(C) }55\qquad\textbf{(D) }60\qquad \textbf{(E) }70$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: Four students took an exam. Three of the scores are $70$, $80$, and $90$. The average of all four scores is $70$. Find the fourth score.

Givens: Three known scores: $70$, $80$, $90$; Total number of students $= 4$; Average of all four scores $= 70$; Answer choices: (A) $40$, (B) $50$, (C) $55$, (D) $60$, (E) $70$

Unknowns: The fourth (missing) score

Understand

Restated: Four students took an exam. Three of the scores are $70$, $80$, and $90$. The average of all four scores is $70$. Find the fourth score.

Givens: Three known scores: $70$, $80$, $90$; Total number of students $= 4$; Average of all four scores $= 70$; Answer choices: (A) $40$, (B) $50$, (C) $55$, (D) $60$, (E) $70$

Plan

Primary tool: #14 Work Backwards

Secondary: #2 Use a Variable

The average is already given, so we can run the mean formula in reverse: instead of dividing a known sum by $4$, multiply the target average by $4$ to recover the required total. That is Tool #14 (Work Backwards). Once we know the total the four scores must add to, the fourth score is just the total minus the three known scores. Tool #2 (Use a Variable) lets us write the missing score as $x$ so the relationship becomes a clean one-line equation.

Execute — Answer: A

#2 Use a Variable 6.EE.B.6 Step 1

Name the unknown.
Let $x$ be the fourth student's score so we can write the average as an equation.

$$x = \text{fourth score}$$

💡 Using a letter for the unknown is the Grade 6 "use variables to represent numbers" move.

#14 Work Backwards 6.SP.B.5 Step 2

Work backwards from the average to the total.
The mean of $4$ numbers equals their sum divided by $4$, so the sum must equal the mean times $4$.

$$\text{sum of all four scores} = 70 \times 4 = 280$$

💡 Reversing "divide by $4$" with "multiply by $4$" is exactly the Tool #14 idea — start from the answer and undo the operation.

#14 Work Backwards 4.NBT.B.4 Step 3

Add the three known scores to find how much of the total is already accounted for.

$$70 + 80 + 90 = 240$$

💡 A straightforward multi-digit addition, well within Grade 4 fluency.

#2 Use a Variable 6.EE.B.7 Step 4

The fourth score is whatever is left after the three known scores fill in.
Subtract to isolate $x$.

$$x = 280 - 240 = 40 \;\Rightarrow\; \textbf{(A)}$$

💡 Solving $240 + x = 280$ in one step is the Grade 6 one-variable equation skill.

[1] #2 6.EE.B.6 Name the unknown. Let $x$ be the fourth student's score so we can write the aver

[2] #14 6.SP.B.5 Work backwards from the average to the total. The mean of $4$ numbers equals the

[3] #14 4.NBT.B.4 Add the three known scores to find how much of the total is already accounted fo

[4] #2 6.EE.B.7 The fourth score is whatever is left after the three known scores fill in. Subtr

Review

Reasonableness: Plug the answer back in: $\dfrac{70 + 80 + 90 + 40}{4} = \dfrac{280}{4} = 70$, which is exactly the given average. The fourth score also lines up with intuition — the three known scores ($70$, $80$, $90$) average to $80$, which is $10$ above the target mean of $70$, so the fourth score must be $30$ below the mean to cancel that $+10 \times 3 = +30$ surplus. That gives $70 - 30 = 40$, matching choice (A).

Alternative: Tool #3 (Look for a Pattern) — balance the deviations from the mean. Compared with the target average $70$: $70$ contributes $0$, $80$ contributes $+10$, $90$ contributes $+20$, total surplus $+30$. The fourth score must contribute $-30$, so it is $70 - 30 = 40$. Same answer (A) without ever computing the sum $280$.

CCSS standards used (min grade 6)

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Adding the three known scores $70 + 80 + 90 = 240$ and subtracting $280 - 240 = 40$.)
6.EE.B.6 Use variables to represent numbers and write expressions when solving real-world problems (Letting $x$ stand for the unknown fourth score so the average relationship can be written as an equation.)
6.EE.B.7 Solve real-world and mathematical problems by writing and solving equations of the form $x + p = q$ (Solving $240 + x = 280$ to get $x = 40$.)
6.SP.B.5 Summarize numerical data sets by giving quantitative measures of center (mean) (Using the definition of the arithmetic mean ($\text{sum} \div \text{count}$) in reverse to recover the required total $70 \times 4 = 280$.)

⭐ Whenever the average is given, multiply it by the count to get the total — then the missing number is just one subtraction away.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 6)

More like this