A boy is now twice as old as his sister, four years ago, he was thrice as old as her what are their ages now?

Finding Boy and Sister's Current Ages

This problem involves solving for the current ages of a boy and his sister based on two conditions relating their ages at different times. We can use algebra to find their ages.

Setting Up Age Variables and Equations

Let's define the variables:

- Let $B$ represent the boy's current age.
- Let $S$ represent the sister's current age.

Now, let's translate the given information into mathematical equations:

1. "A boy is now twice as old as his sister." This translates to the equation: $$B=2S$$
2. "four years ago, he was thrice as old as her." First, express their ages four years ago: - Boy's age 4 years ago: $B-4$ - Sister's age 4 years ago: $S-4$ The condition states the boy's age then was three times the sister's age then: $$B-4=3(S-4)$$

Solving the System of Age Equations

We have a system of two linear equations:

1. $B=2S$
2. $B-4=3(S-4)$

We can solve this system using substitution. Substitute the expression for $B$ from the first equation into the second equation:

$$(2S)-4=3(S-4)$$

Now, simplify and solve for $S$:

First, distribute the 3 on the right side:

$$2S-4=3S-12$$

Next, gather the $S$ terms on one side and the constants on the other. Subtract $2S$ from both sides:

$$-4=S-12$$

Add 12 to both sides:

$$-4+12=S$$

$$8=S$$

So, the sister's current age is 8 years.

Calculating the Boy's Current Age

Use the first equation ($B=2S$) to find the boy's current age:

$$B=2\times S$$

$$B=2\times 8$$

$$B=16$$

The boy's current age is 16 years.

Verifying the Ages

Let's check if these ages (Boy = 16, Sister = 8) satisfy both conditions:

- Condition 1 Check: Is the boy currently twice as old as his sister? $16=2\times 8$. This is true.
- Condition 2 Check: Four years ago, the boy was $16-4=12$ years old, and the sister was $8-4=4$ years old. Was the boy thrice as old as his sister? $12=3\times 4$. This is also true.

Both conditions are met with the ages 16 and 8.

Conclusion on Ages

The current ages are:

- Boy: 16 years
- Sister: 8 years

These ages match the option (16, 8).