Question 1
1. The reciprocal of 5 plus the reciprocal of 7 is the reciprocal of what number?
Answer:
The reciprocal of 5 plus the reciprocal of 7 is the reciprocal of [tex]\frac{12}{35}[/tex]
Solution:
1. The reciprocal of 5 plus the reciprocal of 7 is the reciprocal of what number?
From given question,
[tex]\text{ reciprocal of 5} = \frac{1}{5}[/tex]
[tex]\text{ reciprocal of 7} = \frac{1}{7}[/tex]
Given that,
reciprocal of 5 + reciprocal of 7 = ?
[tex]\frac{1}{5} + \frac{1}{7} = x[/tex]
On cross-multiplying we get,
[tex]\frac{1}{5} + \frac{1}{7} = \frac{7+5}{5 \times 7} = \frac{12}{35}[/tex]
Thus reciprocal is [tex]\frac{35}{12}[/tex]
So the reciprocal of 5 plus the reciprocal of 7 is the reciprocal of [tex]\frac{12}{35}[/tex]
Question 2
2. The reciprocal of the product of two consecutive integers is 1/72
Answer:
The value of two consecutive numbers are 8 and 9
Solution:
Let the two consecutive integers be x and x + 1
Given that reciprocal of product of two consecutive integers is [tex]\frac{1}{72}[/tex]
product of two consecutive integers = x(x + 1) = [tex]x^2 + x[/tex]
reciprocal of the product of two consecutive integers = [tex]\frac{1}{72}[/tex]
[tex]\frac{1}{x^2 + x} = \frac{1}{72}\\\\x^2 + x = 72\\\\x^2 + x - 72 = 0[/tex]
Solve the above quadratic equation by grouping method
[tex]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[/tex]
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
