Question

The average age of Elliot, Miya and Cara is 4 years more than the average

age of Elliot and Miya.
Cara is 9 years older than Elliot.
Four years ago, Elliot was double the age of Miya.
Work out the ages of Elliot, Miya and Cara.

Answer 1

Elliot = 16 years old

Miya = 10 years old

Cara = 25 years old

Explanation:

Define the variables:

• x = Elliot's age
• y = Miya's age
• z = Cara's age

The average age of Elliot, Miya and Cara is 4 years more than the average age of Elliot and Miya:

`\implies (x+y+z)/(3)=(x+y)/(2)+4`

`\implies (x+y+z)/(3)=(x+y+8)/(2)`

`\implies 2(x+y+z)=3(x+y+8)`

`\implies 2x+2y+2z=3x+3y+24`

`\implies 2z=x+y+24`

`\implies x=2z-y-24`

Cara is 9 years older than Elliot:

`\implies z = x + 9`

Four years ago, Elliot was double the age of Miya:

`\implies (x - 4) = 2(y - 4)`

`\implies x - 4 = 2y-8`

`\implies x +4 = 2y`

`\implies y=(1)/(2)x +2`

Substitute the equations for z and y into the equation for x and solve for x:

`\implies x=2z-y-24`

`\implies x=2(x+9)-\left((1)/(2)x +2\right)-24`

`\implies x=(3)/(2)x-8`

`\implies (1)/(2)x=8`

`\implies x=16`

Substitute the found value of x into the equation for z and solve for z:

`\implies z = 16 + 9`

`\implies z = 25`

Substitute the found value of x into the equation for y and solve for y:

`\implies y=(1)/(2)(16) +2`

`\implies y=8+2`

`\implies y=10`

Therefore, the ages of Elliot, Miya and Cara are:

• Elliot = 16 years old
• Miya = 10 years old
• Cara = 25 years old