Julia is 4 years older than twice Kelly's age. The product of their ages is 96. Determine Kelly's age algebraically.

Answer 1

FIND: Kelly's age algebraically

GIVEN:

Julia's age = 4 + 2 * Kelly's age

Julia's age * Kelly's age = 96

Set K = Kelly's age

Set J = Julia's age

Algebracically:

J = 4 + 2 K

J*K = 96

We can solve for Kelly's age since we have 2 equations and 2 unknowns as follows.

Combine both equations into one equation with one unknown:

(4 + 2K)K = 96

Solve for K

4K + 2K² = 96

2K² + 4K -96 = 0

Simplifying

K² + 2K -48 = 0

Factoring:

(K-6)(K+8) = 0

Solving for K:

K-6 = 0

K = 6

SOLUTION:

Kelly is 6 years old

CHECKING RESULT:

J = 4 +2*K

J = 4 + 2*6

J = 16

J*K = 96

16*6 = 96

96 = 96

Since both sides of the equation are identical, the result verified

Answer 2

Given Information

We were provided this information in the text.

"Julia is 4 years older than twice Kelly's age. The product of their ages is 96. Determine Kelly's age algebraically."

Rewritten in equation format

We retell the given information with numbers and titles.

(Julia's Age) = (2 * Kelly's Age) + 4

(Julia's Age) * (Kelly's Age) = 96

Assign a letter for each unknown age.

These letters will serve as our variables.

"Julia's Age" = J

"Kelly's Age" = K

Equation rewritten with variables

This will make the equations easier to see in mathematical terms.

J = 2K + 4

JK = 96

Solving Phase: Part 1 - Combining both equations by isolating J for both.

If both equations equal J, then both equations equal each other. We no longer have to solve for J and will only have one variable.

Equation 1 (J is isolated, complete): J = 2k + 4

Equation 2: JK = 96. Isolate J by dividing both sides of the equation by K.

Equation 2 (J is isolated, complete) J = 96/K

Combined Equation: 2K + 4 = 96/K

Since it's easier to solve with a common denominator, we will multiply both sides of the equation by "K" so both parts have the same denominator.

Combined Equation with the Common Denominator: 2K² + 4K = 96

Solving Phase: Part 2 - Restructure the remaining equation by setting all contents equal to zero.

Use this restructuring to factor the polynomial - This is how we can determine what K could be.

Combined Equation with the Common Denominator: 2K² + 4K = 96

Equation set equal to zero: 2K² + 4K - 96 = 0.

2 is the Greatest Common Factor of all the parts of the equation, so we can divide everything by 2 to simplify all parts of the equation: K² + 2K - 48 = 0.

We can factor this equation by grouping.

Which two numbers will add up to 2, and their product will be -48?

48 is the product of any of the below combinations. However, since there is a negative in our equation (-48), one of the two numbers has to be negative. And, since their sum is a positive number, that means the larger of the two numbers has to be the one that is positive.

1. -1*48 : Their sum is 47. Incorrect.
2. -2*24 : Their sum is 22. Incorrect.
3. -3*16 : Their sum is 13. Incorrect.
4. -4*12 : Their sum is 8. Incorrect.
5. -6*8 : Their sum is 2. This is the only applicable option.

Using "-6" and "8," we can create our factor groups.

Equation in factored groups: (K-6)*(K+8) = 0.

Solving Phase: Part 3 - Factor the polynomial.

Since anything multiplied by 0 will equal 0, we have to identify what value(s) make each section of the factored polynomial equal 0.

(K-6) = 0 if K is 6.

(K+8) = 0 if K is -8.

Interpret the values of K and answer the question.

We were asked to determine an age, so that's what our variables represented. Ages cannot be negative, so negative values are disallowed as an answer.

K is either 6 or -8. K represents "Kelly's Age," and ages cannot be negative. Therefore, only 6 is a viable answer.

Conclusion: Kelly's age is 6 years old.

If you would like to verify & double check this answer, plug "6" in for "K" to our previous equations.

"J = 2K + 4" becomes "J = (2*6) +4 = 16."

"JK = 96" becomes "J*6 = 96," and when J is isolated, it becomes "J = 16."

The findings are consistent in every application. We are certain that we are correct.

Answer 3

This can be solved using systems of linear equations and quadratics. I will use j for Julia and k for Kelly.

What's given is that Julia is twice Kelly's age (we can express as 2k) + 4:

j = 2k + 4

And that the product of their ages is 96, so we can express that as:

k * j = 96

Since we already know the expression that would give us the value of j (1st system), we can substitute that in for the value of j in the second equation so that we only have k terms:

k(2k + 4) = 96 --> 2k² + 4k = 96 ** divide everything by two (2) to have the leading coefficient = 1**

k² + 2k = 48 (hopefully this looks familiar; it is indeed a quadratic function!)

Next we want to have a perfect square on the left side so that solving for k is a lot less hairy. To find the constant term that would make the left side of the equation a perfect square, we divide b (2) by 2 and then square that answer, which is just 1 since 2 ÷ 2 = 1 and 1² also = 1. Since we're adding 1 on the left side, we also have to add 1 to the right side to make sure we honor the equation:

k² + 2k + 1 = 48 + 1 -----> k² + 2k + 1 = 49 -----> (k+1)² = 49 ** take square root of each side and isolate k **

Note: The square root of 49 does evaluate to + - 7, but since we are speaking in practical terms i.e. age can not take on a negative value, we will only focus on the principal root; positive 7.

k+1 = 7 ----> k = 6

To find Julia's age, we simply plug in the value for Kelly's age since we know j = 2k + 4:

j = 2(6) + 4 ----> 16

if it works for one system, it has to work for the other, so:

16 * 6 = 96

Thus, Julia is 16 and Kelly is 6.

Answer 4

I have solved the problem in this video. Check it out and let me know if you have any questions. I hope this helps. Take care!

Dr. Christal-Joy Turner

Answer 5

As Julies age is given in relation to Kelly, let's define k = Kelly's age.

Then Julie's age is 2k+4

and as the product of the ages is 96, k (2k+4) = 96

k (2k+4) = 96

We can factor 2 out of the parenthesis.

2k+4 = 2(k+2), so 2k (k+2) = 96

Divide by 2, k(k+2) = 48

Think about positive factors of 48

48 = 1 x 48, 2 x 24, 3 x 16, 4 x 12, 6 x 8 , ... so clearly k=6 works here, so Kelly is 6 and Julie is 6 x 2 + 4 = 16.

Alternatively k (k+2)= 48

k^2 + 2k = 48

k^2 + 2k - 48 = 0

(k-6) (k+8) = 0

k-6 = 0 giving k=6

k+8=0 giving k=-8, which clearly cannot work as k >0.

Answer 6

This can be solved by making sure each person's age is clearly defined based on Kelly's age.

(Julia's age) * (Kelly's age) = 96

(Julia's age) = 4 +2x

(Kelly's age) = x

so therefore an equation that can be used to solve for Kelly's age, and then deduce Julia's age is...

x(4 + 2x) = 96

distributive property would result in...

4x + 2x² = 96

We would then need to solve for the zeros in the equation by setting it equal to 0...

4x + 2x² - 96 = 0

2(x² + 2x - 48) = 0

2[(x - 6 )(x + 8 )] = 0

The zeros would be 6 and 8, and now lets look back at the problem to see which would satisfy the conditions of the problem.

If Kelly is 6, then Julia is 2(6) + 4 = 12+4 = 16. Therefore, the product of their ages would be 6 * 16= 96 which checks out as the correct answer.

If kelly is 8, then Julia is 2(8) + 4 = 16+4 = 20. Therefore, the product of their ages would be 8 * 20 = 160 which does NOT check out as the correct answer.

So Kelly is 6 years old, which makes Julia 16.

Answer 7

Read the word problem and identify all of the numerical information given to us that we can use to solve an equation and find out how old Julia and Kelly are.

Start out by naming your vaiables (unknown numbers). What is the question asking us to find?

It's asking us to find out the ages of Julia and Kelly. So, we DON'T KNOW how old they are in years. Those are our 2 variables. I am going to name my variables as follows:

j = Julia's age

k = Kelly's age

They tell us 2 key pieces of information about their ages that we can use to write equations.

1. Julia is 4 years older than Kelly, so instead of using 2 variables (j and k) we can use just one.

julia's age (j) is equal to kelly's age (k) PLUS 4

j = k + 4

The 2nd piece of information they gave us that we can use to solve is - the product of their 2 ages is equal to 96. So.....

j x k = 96

We just said that j - k + 4, so substitute that in the equatioin above

(k + 4) x k = 96

Solve for k to get Kellys Age

k² + 4k = 96

-96 - 96

k² + 4k - 96 = 0

Now solve the quadratic by factoring.

(k + 12)(k - 8) = 0

k can either equal -12 or 8

her age can only make sense as a positive number, so Kelly is 8 years old.

Use that info to find Julia's age.

j = k + 4

j = (8) + 4 = 12 years old

Julia is 12 yrs old and Kelly is 8 yrs old

Answer 8

x(2x + 4) = 96; x² + 2x - 48 = 0; x = 6 years Kelly's age