Question

I am currently 13 times as old as my granddaughter. Next year, I will be 11 times as old as my granddaughter. How old am I now?

Answer 1

Given statements:

• I am currently 13 times as old as my granddaughter.
• Next year, I will be 11 times as old as my granddaughter.

Let us represent the character's age by variables. Assuming that the grandparent's age is G and the granddaughter's age is D. Now, using the obtained variables and the statements, create two equations that represents the provided statements. Then, we can compare the equations and simplify if needed to determine the value of the variables.

`\implies \left[\begin{array}{ccc} \text{G = 13D} \\ \text{G + 1 = 11(D + 1)}\\\end{array}\right]`

Since G = 13D is in the form of direct variation, we can substitute the value of G into the second equation to determine the value of D (the age of the granddaughter). Then, we can substitute the value of D into any of the two equations to determine the age of the grandparent.

• ⇒ G + 1 = 11(D + 1)
• ⇒ G + 1 = 11(D + 1) [G = 13D]
• ⇒ 13D + 1 = 11(D + 1)

To solve for D, simplify both sides of the equation by using the distributive property. We need to simplify both sides of the equation so that we can isolate the variables on the other side of the equation.

• ⇒ 13D + 1 = 11(D + 1)
• ⇒ 13D + 1 = 11D + 11

Isolate the variables on one side. This step is necessary because it can cancel out the extra variable and simplify the other variable. To do this, you can either add or subtract on both sides of the equation.

Note: I am subtracting 11D on both sides of the equation since it is my preference. But you can also subtract 13D on both sides of the equation.

• ⇒ 13D + 1 - 11D = 11D + 11 - 11D
• ⇒ 1 + 2D = 11

To further isolate 2D, we need to subtract 1 on both sides of the equation to cancel out the 1 on the left-hand side of the equation. Then, we have:

• ⇒ 1 + 2D - 1 = 11 - 1
• ⇒ 2D = 10

But the variable is still not completely isolated since there is a coefficient being multiplied to D. To isolate the coefficient from the variable, we will need to divide the coefficient (2) on both sides of the equation.

• ⇒ 2D = 10
• ⇒ 2D/2 = 10/2

Then, simplify both sides of the equation. Eventually, we will have a variable on one side and a numerical value on the other side.

• ⇒ 2D/2 = 10/2
• ⇒ D = 5

Now we know the age of the granddaughter, we can substitute the age of the granddaughter into one of the equations and simplify it to determine the age of the grandparent. I am going to choose G = 13D since it is easier to simplify. Then, we will substitute the value of D into the equation.

• ⇒ G = 13D
• ⇒ G = 13(5)
• ⇒ G = 65 years

Therefore, the age of the grandparent is 65 years.

Answer 2

65 years old.

Explanation:

Define the variables:

• Let x be the age of the granddaughter.
• Let y be the age of the grandparent.

Given:

• I am currently 13 times as old as my granddaughter.
• Next year, I will be 11 times as old as my granddaughter.

Create two equations with the given information and defined variables:

`\begin{cases}y = 13x\\y + 1 = 11(x + 1)\end{cases}`

Substitute the first equation into the second equation and solve for x:

`\implies 13x+1=11(x+1)`

`\implies 13x+1=11x+11`

`\implies 2x=10`

`\implies x=5`

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

`\implies y=13(5)`

`\implies y=65`

Therefore, the age of the grandparent now is 65 years.