The sum of three different numbers is 223. The largest number is 13 more than the smallest number. The sum of the largest number and the smallest number is 7 more than twice the middle number. What is the largest of these numbers?

82.

Step-by-step explanation:

Let the largest number of be 'a'.

Let the middle number be 'b'.

Let the smallest number be 'c'.

From the question given above,

The sum of the three numbers is 223. This can be written as:

a + b + c = 223 (1)

The largest number is 13 more than the smallest number. This can be written as:

a = c + 13 (2)

The sum of the largest number and the smallest number is 7 more than twice the middle number. This can be written as:

a + c = 2b + 7 (3)

Summary:

a + b + c = 223 (1)

a = c + 13 (2)

a + c = 2b + 7 (3)

Substitute the value of a in equation 2 into equation 3. This is illustrated below:

a + c = 2b + 7

a = c + 13

c + 13 + c = 2b + 7

2c + 13 = 2b + 7

Make b the subject

2c + 13 – 7 = 2b

2c + 6 = 2b

2(c + 3) = 2b

Divide both side by 2

b = 2(c + 3) / 2

b = c + 3 (4)

Substitute the value of a in equation 2 and the value of b in equation 4 into equation 1. This is illustrated below:

a + b + c = 223

a = c + 13

b = c + 3

(c + 13) + (c + 3) + c = 223

c + 13 + c + 3 + c = 223

3c + 16 = 223

Collect like terms

3c = 223 – 16

3c = 207

Divide both side by 3

c = 207/3

c = 69

The smallest number (c) is 69

Substitute the value of c into equation 2 to obtain the largest number (a). This is illustrated below:

a = c + 13

c = 69

a = 69 + 13

a = 82

Therefore, the largest number (a) is 82