Type The Correct Answer In The Box. Use Numerals Instead Of Words.Jon Is 3 Years Younger Than Laura. The Product Of Their Ages Is 1,330. If $j$ Represents Jon's Age And $j+3$ Represents Laura's Age, What Value Of

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Introduction

In this problem, we are given two pieces of information about the ages of Jon and Laura. We know that Jon is 3 years younger than Laura, and the product of their ages is 1,330. We are asked to find the value of Jon's age, represented by the variable $j$, and Laura's age, represented by the variable $j+3$. This problem requires us to use algebraic equations to solve for the unknown values.

Understanding the Problem

Let's break down the information given in the problem. We know that Jon is 3 years younger than Laura, which means that Laura's age is 3 years more than Jon's age. We can represent this relationship using the equation:

j+3=j+3j+3 = j+3

This equation states that Laura's age is equal to Jon's age plus 3.

The Product of Their Ages

We are also given that the product of their ages is 1,330. This means that the product of Jon's age and Laura's age is equal to 1,330. We can represent this relationship using the equation:

j(j+3)=1330j(j+3) = 1330

This equation states that the product of Jon's age and Laura's age is equal to 1,330.

Solving the Equation

To solve for the value of $j$, we can start by expanding the equation:

j2+3j=1330j^2 + 3j = 1330

Next, we can rearrange the equation to form a quadratic equation:

j2+3j−1330=0j^2 + 3j - 1330 = 0

Factoring the Quadratic Equation

To solve for the value of $j$, we can try to factor the quadratic equation. Unfortunately, this equation does not factor easily, so we will need to use other methods to solve for the value of $j$.

Using the Quadratic Formula

One method for solving quadratic equations is to use the quadratic formula:

j=−b±b2−4ac2aj = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, the values of $a$, $b$, and $c$ are:

a=1a = 1

b=3b = 3

c=−1330c = -1330

Substituting these values into the quadratic formula, we get:

j=−3±32−4(1)(−1330)2(1)j = \frac{-3 \pm \sqrt{3^2 - 4(1)(-1330)}}{2(1)}

Simplifying the Expression

Simplifying the expression under the square root, we get:

j=−3±9+53202j = \frac{-3 \pm \sqrt{9 + 5320}}{2}

j=−3±53292j = \frac{-3 \pm \sqrt{5329}}{2}

Evaluating the Square Root

Evaluating the square root, we get:

j=−3±732j = \frac{-3 \pm 73}{2}

Solving for the Value of $j$

We now have two possible values for the value of $j$:

j=−3+732j = \frac{-3 + 73}{2}

j=−3−732j = \frac{-3 - 73}{2}

Simplifying these expressions, we get:

j=702j = \frac{70}{2}

j=−762j = \frac{-76}{2}

j=35j = 35

j=−38j = -38

Checking the Solutions

We now have two possible values for the value of $j$: 35 and -38. We can check these solutions by plugging them back into the original equation:

j(j+3)=1330j(j+3) = 1330

For $j = 35$, we get:

35(35+3)=35(38)=133035(35+3) = 35(38) = 1330

This solution checks out.

For $j = -38$, we get:

−38(−38+3)=−38(−35)=−1330-38(-38+3) = -38(-35) = -1330

This solution does not check out.

Conclusion

We have found that the value of $j$ is 35. This means that Jon's age is 35, and Laura's age is 35 + 3 = 38. Therefore, the value of $j$ is 35.

Answer

The value of $j$ is 35.

Introduction

In our previous article, we solved the age problem by using algebraic equations to find the value of Jon's age, represented by the variable $j$, and Laura's age, represented by the variable $j+3$. We found that the value of $j$ is 35, which means that Jon's age is 35 and Laura's age is 35 + 3 = 38.

Q&A

Q: What is the relationship between Jon's age and Laura's age?

A: Jon is 3 years younger than Laura.

Q: What is the product of their ages?

A: The product of their ages is 1,330.

Q: How do we represent the relationship between Jon's age and Laura's age using an equation?

A: We can represent this relationship using the equation:

j+3=j+3j+3 = j+3

This equation states that Laura's age is equal to Jon's age plus 3.

Q: How do we represent the product of their ages using an equation?

A: We can represent this relationship using the equation:

j(j+3)=1330j(j+3) = 1330

This equation states that the product of Jon's age and Laura's age is equal to 1,330.

Q: How do we solve for the value of $j$?

A: We can start by expanding the equation:

j2+3j=1330j^2 + 3j = 1330

Next, we can rearrange the equation to form a quadratic equation:

j2+3j−1330=0j^2 + 3j - 1330 = 0

We can then use the quadratic formula to solve for the value of $j$:

j=−b±b2−4ac2aj = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What are the values of $a$, $b$, and $c$ in the quadratic equation?

A: The values of $a$, $b$, and $c$ are:

a=1a = 1

b=3b = 3

c=−1330c = -1330

Q: How do we simplify the expression under the square root?

A: We can simplify the expression under the square root by evaluating the square root:

j=−3±53292j = \frac{-3 \pm \sqrt{5329}}{2}

Q: What are the two possible values for the value of $j$?

A: We have two possible values for the value of $j$:

j=−3+732j = \frac{-3 + 73}{2}

j=−3−732j = \frac{-3 - 73}{2}

Simplifying these expressions, we get:

j=702j = \frac{70}{2}

j=−762j = \frac{-76}{2}

j=35j = 35

j=−38j = -38

Q: How do we check the solutions?

A: We can check the solutions by plugging them back into the original equation:

j(j+3)=1330j(j+3) = 1330

For $j = 35$, we get:

35(35+3)=35(38)=133035(35+3) = 35(38) = 1330

This solution checks out.

For $j = -38$, we get:

−38(−38+3)=−38(−35)=−1330-38(-38+3) = -38(-35) = -1330

This solution does not check out.

Conclusion

We have answered some of the most frequently asked questions about the age problem. We have found that the value of $j$ is 35, which means that Jon's age is 35 and Laura's age is 35 + 3 = 38.

Answer

The value of $j$ is 35.