HELP and need explanation :(

Question

- Artesia is as old as the combined age of her two brothers Amuro and Char. But Char is two years older than Amuro. The combined age of the three last year was ¾ their combined ages at present. How old is Char now?

 - The sum of the digits of a two-digit number is 11. If the digits are interchanged, the new number is 27 larger than the original number.

Answer 1

Hi Chaos,

 

This will come out to a system of 3 equations with 3 unknowns.

Those can be a bit unwieldy, but if you take your time and follow

the logic trails, they aren't that bad.

 

First, pick variables to represent the unknowns

Let x = Artesia's age

Let y = Amuro's age

Let z = Char's age

 

Make equations from the information given:

 

Artesia is as old as the combined age of her two brothers:

x = y + z      {equation 1}

 

Char is two years older than Amuro:

z = y + 2      {equation 2}

 

The combined age of the three last year was ¾ their combined ages at present:

Note:  Last year their ages were.. x-1, y-1, z-1

(x-1)+(y-1)+(z-1) = (3/4)(x+y+z)

 

Let's simplify this down

 

x + y + z - 3 = (3/4)(x+y+z)

 

Clear the fraction by multiplying through by 4

 

4x + 4y + 4z - 12 = 3(x+y+z)

4x + 4y + 4z - 12 = 3x + 3y + 3z

 

Subtract (3x+3y+3z) from both sides

 

x + y + z - 12 = 0

x + y + z = 12       {equation 3}

 

Let's look at what we have so far:

 

x = y + z            {equation 1}

z = y + 2            {equation 2}

x + y + z = 12    {equation 3}

 

We want to find Char's age now ... that is "z"

From equation 1:

y = x - z

From equation 2:

y = z - 2

so

x - z = z - 2

x = 2z - 2

 

So, we have x in terms of z and y in terms of z..

Substitute 2z-2 in place of x and z-2 in place of y in equation 3

 

(2z-2)+(z-2)+z = 12

4z - 4 = 12

4z = 16

z = 4

Char is 4 years old...

 

y=z-2 = 2 years old for Amuro

 

x = 2z-2 = 6 years old for Artesia

 

If you plug those 3 values into the 3 equations you will find they add up correctly.

Answer 2

C represents Char's age

C - 2 represents Amuro's age

C + C - 2 represents Artesia's age

4C - 4 represent their combined age

4C - 4 - 3 represents the combined age last year

4C - 7 = 0.75(4C - 4)

 

Can you solve for C and answer?