A sum of 700 is to be paid to give seven cash prizes to the students of a school for their overall academic performance. If the cost of each prize is 20 less than its preceding prize, find the value of each of the prizes.

Answer

To solve the problem step by step, we will use the concept of Arithmetic Progression (AP).

### Step 1: Define the first prize
Let the value of the first prize be \( A \).

### Step 2: Define the subsequent prizes
According to the problem, each subsequent prize is 20 less than the preceding one. Therefore, the prizes can be expressed as:
- First prize: \( A \)
- Second prize: \( A - 20 \)
- Third prize: \( A - 40 \)
- Fourth prize: \( A - 60 \)
- Fifth prize: \( A - 80 \)
- Sixth prize: \( A - 100 \)
- Seventh prize: \( A - 120 \)

### Step 3: Write the sum of the prizes
The total sum of the prizes is given as 700. Therefore, we can write the equation for the sum of the prizes:
\[ A + (A - 20) + (A - 40) + (A - 60) + (A - 80) + (A - 100) + (A - 120) = 700 \]

### Step 4: Simplify the equation
Combining like terms, we have:
\[ 7A - (20 + 40 + 60 + 80 + 100 + 120) = 700 \]
Calculating the sum of the constants:
\[ 20 + 40 + 60 + 80 + 100 + 120 = 420 \]
So, the equation becomes:
\[ 7A - 420 = 700 \]

### Step 5: Solve for \( A \)
Now, we can isolate \( A \):
\[ 7A = 700 + 420 \]
\[ 7A = 1120 \]
\[ A = \frac{1120}{7} = 160 \]

### Step 6: Calculate the values of all prizes
Now that we have \( A = 160 \), we can find the values of all the prizes:
- First prize: \( 160 \)
- Second prize: \( 160 - 20 = 140 \)
- Third prize: \( 160 - 40 = 120 \)
- Fourth prize: \( 160 - 60 = 100 \)
- Fifth prize: \( 160 - 80 = 80 \)
- Sixth prize: \( 160 - 100 = 60 \)
- Seventh prize: \( 160 - 120 = 40 \)

### Final Values of the Prizes
The values of the seven prizes are:
- \( 160, 140, 120, 100, 80, 60, 40 \)