<h3><u>Question 1</u></h3>
1. The reciprocal of 5 plus the reciprocal of 7 is the reciprocal of what number?
<h3><u>Answer:</u></h3>
The reciprocal of 5 plus the reciprocal of 7 is the reciprocal of ![\frac{12}{35}](https://tex.z-dn.net/?f=%5Cfrac%7B12%7D%7B35%7D)
<h3><u>Solution:</u></h3>
1. The reciprocal of 5 plus the reciprocal of 7 is the reciprocal of what number?
From given question,
![\text{ reciprocal of 5} = \frac{1}{5}](https://tex.z-dn.net/?f=%5Ctext%7B%20reciprocal%20of%205%7D%20%3D%20%5Cfrac%7B1%7D%7B5%7D)
![\text{ reciprocal of 7} = \frac{1}{7}](https://tex.z-dn.net/?f=%5Ctext%7B%20reciprocal%20of%207%7D%20%3D%20%5Cfrac%7B1%7D%7B7%7D)
Given that,
reciprocal of 5 + reciprocal of 7 = ?
![\frac{1}{5} + \frac{1}{7} = x](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B5%7D%20%2B%20%5Cfrac%7B1%7D%7B7%7D%20%3D%20x)
On cross-multiplying we get,
![\frac{1}{5} + \frac{1}{7} = \frac{7+5}{5 \times 7} = \frac{12}{35}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B5%7D%20%2B%20%5Cfrac%7B1%7D%7B7%7D%20%3D%20%5Cfrac%7B7%2B5%7D%7B5%20%5Ctimes%207%7D%20%3D%20%5Cfrac%7B12%7D%7B35%7D)
Thus reciprocal is ![\frac{35}{12}](https://tex.z-dn.net/?f=%5Cfrac%7B35%7D%7B12%7D)
So the reciprocal of 5 plus the reciprocal of 7 is the reciprocal of ![\frac{12}{35}](https://tex.z-dn.net/?f=%5Cfrac%7B12%7D%7B35%7D)
<h3><u>Question 2</u></h3>
2. The reciprocal of the product of two consecutive integers is 1/72
<h3><u>Answer:</u></h3>
The value of two consecutive numbers are 8 and 9
<h3><u>Solution:</u></h3>
Let the two consecutive integers be x and x + 1
Given that reciprocal of product of two consecutive integers is ![\frac{1}{72}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B72%7D)
product of two consecutive integers = x(x + 1) = ![x^2 + x](https://tex.z-dn.net/?f=x%5E2%20%2B%20x)
reciprocal of the product of two consecutive integers = ![\frac{1}{72}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B72%7D)
![\frac{1}{x^2 + x} = \frac{1}{72}\\\\x^2 + x = 72\\\\x^2 + x - 72 = 0](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bx%5E2%20%2B%20x%7D%20%3D%20%5Cfrac%7B1%7D%7B72%7D%5C%5C%5C%5Cx%5E2%20%2B%20x%20%3D%2072%5C%5C%5C%5Cx%5E2%20%2B%20x%20-%2072%20%3D%200)
Solve the above quadratic equation by grouping method
![x^2 + x - 72 = 0\\\\x^2 -8x + 9x - 72 = 0\\\\x^2 + 9x + (-8x - 72) = 0\\\\x(x + 9) -8(x + 9) = 0\\\\(x + 9)(x - 8) = 0](https://tex.z-dn.net/?f=x%5E2%20%2B%20x%20-%2072%20%3D%200%5C%5C%5C%5Cx%5E2%20-8x%20%2B%209x%20-%2072%20%3D%200%5C%5C%5C%5Cx%5E2%20%2B%209x%20%2B%20%28-8x%20-%2072%29%20%3D%200%5C%5C%5C%5Cx%28x%20%2B%209%29%20-8%28x%20%2B%209%29%20%3D%200%5C%5C%5C%5C%28x%20%2B%209%29%28x%20-%208%29%20%3D%200)
Thus x = -9 or 8
Ignoring negative value,
x = 8
Thus two consecutive integers are x = 8 and x + 1 = 8 + 1 = 9
8 and 9 are two consecutive integers