Math / Algebra

QuestionPat is 20 years older than his son James. In two years Pat will be twice as old as James. How old are they now?

Studdy Solution

STEP 1

What is this asking? How old are Pat and James right now, given their age difference and a future age ratio? Watch out! Don't mix up present and future ages!
Keep track of who's who!

STEP 2

1. Set up variables
2. Create equations
3. Solve for James's age
4. Solve for Pat's age

STEP 3

Let's define some variables to make things easier!
Let $J$ be James's current age and $P$ be Pat's current age.
This will help us keep everything organized!

STEP 4

We know Pat is $\textbf{20}$ years older than James right now.
So, we can write that as an equation: $P = J + 20$ .
See how neat that is?

STEP 5

In $\textbf{two}$ years, Pat's age will be $P + 2$ and James's age will be $J + 2$ .
At that time, Pat will be twice as old as James.
So, another equation pops up: $P + 2 = 2 \cdot (J + 2)$ .
Awesome!

STEP 6

Now, let's substitute the first equation ( $P = J + 20$ ) into the second equation: $(J + 20) + 2 = 2 \cdot (J + 2)$ .
This way, we get an equation with only $J$ , which we can solve!

STEP 7

Let's simplify things: $J + 22 = 2J + 4$ .
Looks much better, right?

STEP 8

Now, subtract $J$ from both sides: $22 = J + 4$ .
Almost there!

STEP 9

Finally, subtract $4$ from both sides to find James's current age: $18 = J$ .
So, James is currently $\textbf{18}$ years old!

STEP 10

Let's use the first equation ( $P = J + 20$ ) again, but this time we know $J = 18$ .
So, $P = 18 + 20$ .

STEP 11

Adding those up, we get $P = 38$ .
Pat is currently $\textbf{38}$ years old!

STEP 12

James is currently $\textbf{18}$ years old, and Pat is currently $\textbf{38}$ years old.

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QuestionPat is 20 years older than his son James. In two years Pat will be twice as old as James. How old are they now?

Studdy Solution

STEP 1

What is this asking? How old are Pat and James right now, given their age difference and a future age ratio? Watch out! Don't mix up present and future ages!
Keep track of who's who!

STEP 2

1. Set up variables
2. Create equations
3. Solve for James's age
4. Solve for Pat's age

STEP 3

Let's define some variables to make things easier!
Let $J$ be James's current age and $P$ be Pat's current age.
This will help us keep everything organized!

STEP 4

We know Pat is $\textbf{20}$ years older than James right now.
So, we can write that as an equation: $P = J + 20$ .
See how neat that is?

STEP 5

In $\textbf{two}$ years, Pat's age will be $P + 2$ and James's age will be $J + 2$ .
At that time, Pat will be twice as old as James.
So, another equation pops up: $P + 2 = 2 \cdot (J + 2)$ .
Awesome!

STEP 6

Now, let's substitute the first equation ( $P = J + 20$ ) into the second equation: $(J + 20) + 2 = 2 \cdot (J + 2)$ .
This way, we get an equation with only $J$ , which we can solve!

STEP 7

Let's simplify things: $J + 22 = 2J + 4$ .
Looks much better, right?

STEP 8

Now, subtract $J$ from both sides: $22 = J + 4$ .
Almost there!

STEP 9

Finally, subtract $4$ from both sides to find James's current age: $18 = J$ .
So, James is currently $\textbf{18}$ years old!

STEP 10

Let's use the first equation ( $P = J + 20$ ) again, but this time we know $J = 18$ .
So, $P = 18 + 20$ .

STEP 11

Adding those up, we get $P = 38$ .
Pat is currently $\textbf{38}$ years old!

STEP 12

James is currently $\textbf{18}$ years old, and Pat is currently $\textbf{38}$ years old.

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QuestionPat is 20 years older than his son James. In two years Pat will be twice as old as James. How old are they now?

Studdy Solution

STEP 1

What is this asking? How old are Pat and James right now, given their age difference and a future age ratio? Watch out! Don't mix up present and future ages!Keep track of who's who!

STEP 2

1. Set up variables2. Create equations3. Solve for James's age4. Solve for Pat's age

STEP 3

Let's **define** some variables to make things easier!Let JJJ be James's **current age** and PPP be Pat's **current age**.This will help us keep everything organized!

STEP 4

We know Pat is 20\textbf{20}20 years older than James *right now*.So, we can write that as an equation: P=J+20P = J + 20P=J+20.See how neat that is?

STEP 5

In two\textbf{two}two years, Pat's age will be P+2P + 2P+2 and James's age will be J+2J + 2J+2.At that time, Pat will be *twice* as old as James.So, another equation pops up: P+2=2⋅(J+2)P + 2 = 2 \cdot (J + 2)P+2=2⋅(J+2).Awesome!

STEP 6

Now, let's **substitute** the first equation (P=J+20P = J + 20P=J+20) into the second equation: (J+20)+2=2⋅(J+2)(J + 20) + 2 = 2 \cdot (J + 2)(J+20)+2=2⋅(J+2).This way, we get an equation with only JJJ, which we can solve!

STEP 7

Let's **simplify** things: J+22=2J+4J + 22 = 2J + 4J+22=2J+4.Looks much better, right?

STEP 8

Now, **subtract** JJJ from both sides: 22=J+422 = J + 422=J+4.Almost there!

STEP 9

Finally, **subtract** 444 from both sides to find James's current age: 18=J18 = J18=J.So, James is currently 18\textbf{18}18 years old!

STEP 10

Let's use the first equation (P=J+20P = J + 20P=J+20) again, but this time we know J=18J = 18J=18.So, P=18+20P = 18 + 20P=18+20.

STEP 11

Adding those up, we get P=38P = 38P=38.Pat is currently 38\textbf{38}38 years old!

STEP 12

James is currently 18\textbf{18}18 years old, and Pat is currently 38\textbf{38}38 years old.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? How old are Pat and James right now, given their age difference and a future age ratio? Watch out! Don't mix up present and future ages!
Keep track of who's who!

1. Set up variables
2. Create equations
3. Solve for James's age
4. Solve for Pat's age

Let's define some variables to make things easier!
Let $J$ be James's current age and $P$ be Pat's current age.
This will help us keep everything organized!

We know Pat is $\textbf{20}$ years older than James right now.
So, we can write that as an equation: $P = J + 20$ .
See how neat that is?

In $\textbf{two}$ years, Pat's age will be $P + 2$ and James's age will be $J + 2$ .
At that time, Pat will be twice as old as James.
So, another equation pops up: $P + 2 = 2 \cdot (J + 2)$ .
Awesome!

Now, let's substitute the first equation ( $P = J + 20$ ) into the second equation: $(J + 20) + 2 = 2 \cdot (J + 2)$ .
This way, we get an equation with only $J$ , which we can solve!

Let's simplify things: $J + 22 = 2J + 4$ .
Looks much better, right?

Now, subtract $J$ from both sides: $22 = J + 4$ .
Almost there!

Finally, subtract $4$ from both sides to find James's current age: $18 = J$ .
So, James is currently $\textbf{18}$ years old!

Let's use the first equation ( $P = J + 20$ ) again, but this time we know $J = 18$ .
So, $P = 18 + 20$ .

Adding those up, we get $P = 38$ .
Pat is currently $\textbf{38}$ years old!

James is currently $\textbf{18}$ years old, and Pat is currently $\textbf{38}$ years old.