3. the sum of the reciprocals of two positive consecutive integers is $\frac{31}{240}$.
a. explain what each of the expressions represents in this problem.
① i. $x$
① ii. $x + 1$
① iii. $\frac{1}{x}$
① iv. $\frac{1}{x + 1}$
① v. $\frac{1}{x} + \frac{1}{x + 1}$
② b. solve the problem. verify the solution.

Question

Accepted Answer

### Part (a)
#### i. $ x $
# Brief Explanations:
$ x $ represents the first of the two positive consecutive integers. Since the integers are consecutive, if the first one is $ x $, the next will be $ x + 1 $.
# Answer:
$ x $ represents the first positive consecutive integer.

#### ii. $ x + 1 $
# Brief Explanations:
Since the integers are consecutive and $ x $ is the first one, $ x + 1 $ represents the second positive consecutive integer (the integer immediately following $ x $).
# Answer:
$ x + 1 $ represents the second positive consecutive integer.

#### iii. $ \frac{1}{x} $
# Brief Explanations:
The reciprocal of a number $ n $ is $ \frac{1}{n} $. Since $ x $ is the first integer, $ \frac{1}{x} $ represents the reciprocal of the first positive consecutive integer.
# Answer:
$ \frac{1}{x} $ represents the reciprocal of the first positive consecutive integer.

#### iv. $ \frac{1}{x + 1} $
# Brief Explanations:
Similarly, $ x + 1 $ is the second integer, so $ \frac{1}{x + 1} $ represents the reciprocal of the second positive consecutive integer.
# Answer:
$ \frac{1}{x + 1} $ represents the reciprocal of the second positive consecutive integer.

#### v. $ \frac{1}{x}+\frac{1}{x + 1} $
# Brief Explanations:
The problem states "the sum of the reciprocals of two positive consecutive integers". $ \frac{1}{x} $ is the reciprocal of the first integer and $ \frac{1}{x + 1} $ is the reciprocal of the second integer, so their sum $ \frac{1}{x}+\frac{1}{x + 1} $ represents the sum of the reciprocals of the two positive consecutive integers.
# Answer:
$ \frac{1}{x}+\frac{1}{x + 1} $ represents the sum of the reciprocals of the two positive consecutive integers.

### Part (b)
# Explanation:
## Step 1: Set up the equation
We know that the sum of the reciprocals is $ \frac{31}{240} $, so we set up the equation:
$$
\frac{1}{x}+\frac{1}{x + 1}=\frac{31}{240}
$$
## Step 2: Find a common denominator
The common denominator of $ x $ and $ x + 1 $ is $ x(x + 1) $. Multiply each term by $ x(x + 1) $ to eliminate the denominators:
$$
x(x + 1)\cdot\frac{1}{x}+x(x + 1)\cdot\frac{1}{x + 1}=x(x + 1)\cdot\frac{31}{240}
$$
Simplify each term:
$$
(x + 1)+x=\frac{31x(x + 1)}{240}
$$
Combine like terms on the left:
$$
2x + 1=\frac{31x(x + 1)}{240}
$$
## Step 3: Multiply both sides by 240 to eliminate the fraction
$$
240(2x + 1)=31x(x + 1)
$$
Expand both sides:
$$
480x+240 = 31x^{2}+31x
$$
## Step 4: Rearrange into standard quadratic form
Subtract $ 480x $ and $ 240 $ from both sides:
$$
31x^{2}+31x-480x - 240=0
$$
Simplify:
$$
31x^{2}-449x - 240=0
$$
## Step 5: Solve the quadratic equation
We can try to factor or use the quadratic formula. Let's use the quadratic formula $ x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} $, where $ a = 31 $, $ b=-449 $, and $ c=-240 $.

First, calculate the discriminant $ D=b^{2}-4ac $:
$$
D=(-449)^{2}-4(31)(-240)
$$
$$
D = 201601+29760
$$
$$
D=231361
$$
Take the square root of $ D $: $ \sqrt{231361}=481 $

Now, find $ x $:
$$
x=\frac{449\pm481}{2\times31}
$$
We have two solutions:
1. $ x=\frac{449 + 481}{62}=\frac{930}{62}=15 $
2. $ x=\frac{449-481}{62}=\frac{-32}{62}=-\frac{16}{31} $

Since we are looking for positive integers, we discard the negative solution. So $ x = 15 $.

## Step 6: Verify the solution
If $ x = 15 $, then the two integers are $ 15 $ and $ 16 $.

Reciprocal of $ 15 $ is $ \frac{1}{15} $, reciprocal of $ 16 $ is $ \frac{1}{16} $.

Sum of reciprocals: $ \frac{1}{15}+\frac{1}{16}=\frac{16 + 15}{240}=\frac{31}{240} $, which matches the given sum. So the solution is correct.
# Answer:
The two positive consecutive integers are $ 15 $ and $ 16 $. Verification: $ \frac{1}{15}+\frac{1}{16}=\frac{16 + 15}{240}=\frac{31}{240} $, which is the given sum.

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QUESTION IMAGE

Question

Explanation:

Part (a)

i. \( x \)

Answer:

ii. \( x + 1 \)