randRange(1, 10) randRangeExclude(1, 10, [I1]) person(I1) person(I2) personVar(I1) personVar(I2)
randRange(3, 5) randRange(2, 20) randRange(1, 10) * (C - 1)

{P1 is `A` years older than P2| P2 is `A` years younger than P1}. {For the last {four|`3`|two} years, P1 and P2 have been friends.| P1 and P2 first met {four|`3`|two} years ago.|} CardinalThrough20(B) years ago, P1 was `C` times as old as P2.

How old is P1 now?

(C * (B + A) - B) / (C - 1)

The information in the first sentence can be expressed in the following equation:

`\blue{V1 = V2 + A}`

$(".first").addClass("hint_blue");

CardinalThrough20(B) years ago, P1 was `V1 - B` years old, and P2 was `V2 - B` years old.

The information in the second sentence can be expressed in the following equation:

`\red{V1 - B = C(V2 - B)}`

$(".second").addClass("hint_red");

Now we have two independent equations, and we can solve for our two unknowns.

Because we are looking for `V1`, it might be easiest to solve our first equation for `V2` and substitute it into our second equation.

Solving our first equation for `V2`, we get: `\blue{V2 = V1 - A}`. Substituting this into our second equation, we get the equation:

`\red{V1 - B = C(} \blue{(V1 - A)}\red{ - B)}`

which combines the information about `V1` from both of our original equations.

Simplifying the right side of this equation, we get: `V1 - B = CV1 - C * (A + B)`.

Solving for `V1`, we get: `C - 1 V1 = C * (A + B) - B`.

`V1 = (C * (B + A) - B) / (C - 1)`.

How old is P2 now?

(A - B + C * B) / (C - 1)

We can use the given information to write down two equations that describe the ages of P1 and P2.

Let P1's current age be `V1` and P2's current age be `V2`.

The information in the first sentence can be expressed in the following equation:

`\blue{V1 = V2 + A}`

$(".first").addClass("hint_blue");

CardinalThrough20(B) years ago, P1 was `V1 - B` years old, and P2 was `V2 - B` years old.

The information in the second sentence can be expressed in the following equation:

`\red{V1 - B = C(V2 - B)}`

$(".second").addClass("hint_red");

Now we have two independent equations, and we can solve for our two unknowns.

Because we are looking for `V2`, it might be easiest to use our first equation for `V1` and substitute it into our second equation.

Our first equation is: `\blue{V1 = V2 + A}`. Substituting this into our second equation, we get the equation:

`\blue{(V2 + A)}\red{-B = C(V2 - B)}`

which combines the information about `V2` from both of our original equations.

Simplifying both sides of this equation, we get: `V2 + A - B = C V2 - C * B`.

Solving for `V2`, we get: `C - 1 V2 = A - B + C * B`.

`V2 = (A - B + C * B) / (C - 1)`.

randRange(3, 5) randRange(2, 10) * (C - 1)

P1 is `C` times as old as P2 and is also `A` years older than P2.

How old is P1?

A * C / (C - 1)

`\blue{V1 = CV2}`

`\red{V1 = V2 + A}`

$(".first").addClass("hint_blue"); $(".second").addClass("hint_red");

Now we have two independent equations, and we can solve for our two unknowns.

One way to solve for `V1` is to solve the second equation for `V2` and substitute that value into the first equation.

Solving our second equation for `V2`, we get: `\red{V2 = V1 - A}`. Substituting this into our first equation, we get the equation:

`\blue{V1 = C}\red{(V1 - A)}`

which combines the information about `V1` from both of our original equations.

Simplifying the right side of this equation, we get: `V1 = CV1 - C * A`.

Solving for `V1`, we get: `C - 1 V1 = A * C`.

`V1 = A * C / (C - 1)`.

How old is P2?

A / (C - 1)

We can use the given information to write down two equations that describe the ages of P1 and P2.

Let P1's current age be `V1` and P2's current age be `V2`.

We can use the given information to write down two equations that describe the ages of P1 and P2.

Let P1's current age be `V1` and P2's current age be `V2`.

`\blue{V1 = CV2}`

`\red{V1 = V2 + A}`

$(".first").addClass("hint_blue"); $(".second").addClass("hint_red");

Now we have two independent equations, and we can solve for our two unknowns.

Since we are looking for `V2`, and both of our equations have `V1` alone on one side, this is a convenient time to use elimination.

Subtracting the second equation from the first equation, we get:

`0 = \blue{CV2} -\red{(V2 + A)}`

which combines the information about `V2` from both of our original equations.

Solving for `V2`, we get: `C - 1 V2 = A`.

`V2 = A / (C - 1)`.

randRange(2, 5) randRange(A + 2, 9) randRange(2, 7) * (C - A)

P1 is `A` times as old as P2. `B` years ago, P1 was `C` times as old as P2.

How old is P1 now?

A * B * (C - 1) / (C - A)

The information in the first sentence can be expressed in the following equation:

`\blue{V1 = AV2}`

$(".first").addClass("hint_blue");

CardinalThrough20(B) years ago, P1 was `V1 - B` years old, and P2 was `V2 - B` years old.

The information in the second sentence can be expressed in the following equation:

`\red{V1 - B = C(V2 - B)}`

$(".second").addClass("hint_red");

Now we have two independent equations, and we can solve for our two unknowns.

Because we are looking for `V1`, it might be easiest to solve our first equation for `V2` and substitute it into our second equation.

Solving our first equation for `V2`, we get: `\blue{V2 = \dfrac{V1}{A}}`. Substituting this into our second equation, we get:

`\red{V1 - B = C (}\blue{\frac{V1}{A}} \red{- B)}`

which combines the information about `V1` from both of our original equations.

Simplifying the right side of this equation, we get: `V1 - B = fractionReduce(C, A) V1 - C * B`.

Solving for `V1`, we get: `fractionReduce(C - A, A) V1 = B * (C - 1)`.

`V1 = fractionReduce(A, C - A) \cdot B * (C - 1) = A * B * (C - 1) / (C - A)`.

How old is P2 now?

B * (C - 1) / (C - A)

We can use the given information to write down two equations that describe the ages of P1 and P2.

Let P1's current age be `V1` and P2's current age be `V2`.

The information in the first sentence can be expressed in the following equation:

`\blue{V1 = AV2}`

$(".first").addClass("hint_blue");

CardinalThrough20(B) years ago, P1 was `V1 - B` years old, and P2 was `V2 - B` years old.

The information in the second sentence can be expressed in the following equation:

`\red{V1 - B = C(V2 - B)}`

$(".second").addClass("hint_red");

Now we have two independent equations, and we can solve for our two unknowns.

Because we are looking for `V2`, it might be easiest to use our first equation for `V1` and substitute it into our second equation.

Our first equation is: `\blue{V1 = AV2}`. Substituting this into our second equation, we get:

`\blue{AV2} \red {-B = C(V2 - B)}`

which combines the information about `V2` from both of our original equations.

Simplifying the right side of this equation, we get: `A V2 - B = C V2 - B * C`.

Solving for `V2`, we get: `C - A V2 = B * (C - 1).`

`V2 = B * (C - 1) / (C - A)`.

randRange(3, 20) randRange(7, 24) * (A - 1)

In `B` years, P1 will be `A` times as old as he is right now.

In `B` years, P1 will be `A` times as old as she is right now.

How old is he right now?

How old is she right now?

B / (A - 1)

We can use the given information to write down an equation about P1's age.

Let P1's age be `V1`.

In `B` years, he will be `V1 + B` years old.

In `B` years, she will be `V1 + B` years old.

At that time, he will also be `A V1` years old.

At that time, she will also be `A V1` years old.

Writing this information as an equation, we get:

`V1 + B = A V1`

Solving for `V1`, we get: `A - 1 V1 = B`.

`V1 = B / (A - 1)`.

randRange(3, 5) randRange(1, 10) * (C - 1) randRange(C * B + 1, 15) * (C - 1)

P1 is `A` years old and P2 is `B` years old.

How many years will it take until P1 is only `C` times as old as P2?

(A - B * C) / (C - 1)

We can use the given information to write down an equation about how many years it will take.

Let `y` be the number of years that it will take.

In `y` years, P1 will be `A + y` years old and P2 will be `B + y` years old.

At that time, P1 will be C times as old as P2.

Writing this information as an equation, we get:

`A + y = C (B + y)`

Simplifying the right side of this equation, we get: `A + y = C * B + C y`.

Solving for `y`, we get: `C - 1 y = A - C * B`.

`y = (A - C * B) / (C - 1)`.

We can use the given information to write down two equations that describe the ages of P1 and P2.

Let P1's current age be `V1` and P2's current age be `V2`.