Question 5

I have a set of four numbers P, Q, R, S in ascending order. The average of P, Q, and R is 22, and the average of Q, R, and 5 is 24. The largest number is 27. Then, the smallest number in this set:

A. is prime

22

B. is a multiple of 9

C. is a multiple of 11

D. is a multiple of 7

E. cannot be determined with this information

Let's break down the information given in the problem step by step to find the smallest number in the set.

You are given four numbers \( P, Q, R, S \) in ascending order:
1. The average of \( P, Q, R \) is 22.
2. The average of \( Q, R, S \) is 24.
3. The largest number \( S \) is 27.

### Step 1: Translate the averages into equations.

From the first condition:
\[
\frac{P + Q + R}{3} = 22
\]
Multiplying both sides by 3, we get:
\[
P + Q + R = 66
\]

From the second condition:
\[
\frac{Q + R + S}{3} = 24
\]
Multiplying both sides by 3, we get:
\[
Q + R + S = 72
\]

### Step 2: Substitute \( S = 27 \) into the second equation.

\[
Q + R + 27 = 72
\]
Subtract 27 from both sides:
\[
Q + R = 45
\]

### Step 3: Use these two equations to find \( P \).

From the first equation, we know:
\[
P + Q + R = 66
\]
We also have \( Q + R = 45 \). Subtract the second equation from the first:
\[
P = 66 - 45 = 21
\]

So, the smallest number \( P \) is 21.

### Step 4: Analyze the answer choices.

- 21 is not a prime number.
- 21 is **not a multiple of 9**.
- 21 is **not a multiple of 11**.
- 21 is **a multiple of 7** (since \( 21 = 7 \times 3 \)).

Thus, the correct answer is:
\[
\boxed{D. \text{is a multiple of 7}}
\]

Would you like further details or have any questions? Here are five related questions:

1. How can you verify if a number is prime or not?
2. What other properties of numbers can be inferred from their factors?
3. How would the solution change if the largest number were different?
4. Can you extend the reasoning to sets of five numbers with given conditions?
5. How do averages relate to the sum of numbers in a set?

**Tip:** When dealing with averages, multiplying both sides of the equation by the number of terms can help transform it into a simple sum equation.

I have a set of four numbers P, Q, R, S in ascending order. The average of P, Q, and R is 22, and the average of Q, R, and 5 is 24. The largest number is 27. Then, the smallest number in this set: A. is prime B. is a multiple of 9 C. is a multiple of 11 D. is a multiple of 7 E. cannot be determined with this information

Discover how to solve a problem involving the average of a set of four numbers P, Q, R, and S in ascending order. Learn how to determine the smallest number, identify multiples, and apply algebraic equations step by step. Suitable for Grades 6-8 students.

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