Question

The red and purple presents together cost £38,

The purple and blue presents together cost £31,

The blue and gold presents together cost £25,

The gold and green presents together cost £23,

The green and red presents together cost £23

What is the total cost of all five presents?

Answer 1

Let's denote the cost of the presents with the following variables:

- \( R \) for the cost of the red present
- \( P \) for the cost of the purple present
- \( B \) for the cost of the blue present
- \( G \) for the cost of the gold present
- \( N \) for the cost of the green present

From the information given, we can form the following equations:

1. \( R + P = 38 \) (Equation 1)
2. \( P + B = 31 \) (Equation 2)
3. \( B + G = 25 \) (Equation 3)
4. \( G + N = 23 \) (Equation 4)
5. \( N + R = 23 \) (Equation 5)

Now, we'll solve these equations step by step.

**Step 1:** Solve for \( R \) in terms of \( P \) using Equation 1:
\[
R = 38 - P
\]

**Step 2:** Solve for \( P \) in terms of \( B \) using Equation 2:
\[
P = 31 - B
\]

**Step 3:** Substitute the expression for \( P \) into the expression for \( R \):
\[
R = 38 - (31 - B) = 38 - 31 + B = 7 + B
\]

**Step 4:** Solve for \( B \) in terms of \( G \) using Equation 3:
\[
B = 25 - G
\]

**Step 5:** Substitute \( B = 25 - G \) into \( R = 7 + B \):
\[
R = 7 + (25 - G) = 32 - G
\]

**Step 6:** Now solve for \( G \) in terms of \( N \) using Equation 4:
\[
G = 23 - N
\]

**Step 7:** Substitute \( G = 23 - N \) into \( R = 32 - G \):
\[
R = 32 - (23 - N) = 32 - 23 + N = 9 + N
\]

**Step 8:** Solve for \( N \) in terms of \( R \) using Equation 5:
\[
N = 23 - R
\]

**Step 9:** Now substitute \( R = 9 + N \) into \( N = 23 - R \):
\[
N = 23 - (9 + N) \\
N + N = 23 - 9 \\
2N = 14 \\
N = 7
\]

**Step 10:** Now substitute \( N = 7 \) back to find \( R \):
\[
R = 9 + 7 = 16
\]

**Step 11:** Now find \( G \) using \( G = 23 - N \):
\[
G = 23 - 7 = 16
\]

**Step 12:** Find \( B \) using \( B = 25 - G \):
\[
B = 25 - 16 = 9
\]

**Step 13:** Find \( P \) using \( P = 31 - B \):
\[
P = 31 - 9 = 22
\]

**Step 14:** Finally, with all present costs calculated, we summarize as follows:
- \( R = 16 \)
- \( P = 22 \)
- \( B = 9 \)
- \( G = 16 \)
- \( N = 7 \)

To find the total cost of all presents, we sum these amounts:
\[
R + P + B + G + N = 16 + 22 + 9 + 16 + 7 = 70
\]

Thus, the total cost of all five presents is \( \boxed{70} \).