I really don't understand this so i would really appreciate if you helped me.
Q. Find two consecutive integers whose reciprocals add up to 7/12.
Thanks x
Two consecutive integers are x and x + 1
1 / x + 1 / ( x + 1 ) = 7 / 12
( x + 1 + x ) / [ x ∙ ( x + 1 ) ] = 7 / 12
( 2 x + 1 ) / ( x² + x ) = 7 / 12
12 ∙ ( 2 x + 1 ) = 7 ∙ ( x² + x )
24 x + 12 = 7 x² + 7 x
0 = 7 x² + 7 x - 24 x - 12
0 = 7 x² - 17 x - 12
7 x² - 17 x - 12 = 0
The solutions are x = - 4 / 7 an x = 3
- 4 / 7 isn't integer so x = 3
Proof:
1 / 3 + 1 / 4 = 4 / 12 + 3 / 12 = 7 / 12
thanks, but i have a question. on the 6th step you move the 7 x² + 7 x to the other side but why do you multiply everything by -1?
When you move some variable on other side of equation you must change sign.
24 x + 12 = 7 x² + 7 x
is same
24 x + 12 - ( 24 x + 12 ) = 7 x² + 7 x - ( 24 x + 12 )
0 = 7 x² + 7 x - ( 24 x + 12 )
0 = 7 x² + 7 x - 24 x - 12
7 x² + 7 x - 24 x - 12 = 0
To solve this problem, we need to set up an equation based on the given information.
Let's assume that the first integer is represented by "x". Since the consecutive integer will be the next number, it can be represented by "x + 1".
According to the problem statement, the reciprocals of these two integers (1/x and 1/(x + 1)) add up to 7/12. Thus, we can write the equation as follows:
1/x + 1/(x + 1) = 7/12
To solve this equation, we need to find a common denominator and then combine the fractions on the left-hand side. The common denominator is x(x + 1), so we can rewrite the equation as:
[(x + 1) + x] / [x(x + 1)] = 7/12
Now, we can simplify the equation:
(2x + 1) / [x(x + 1)] = 7/12
Cross-multiplying:
12(2x + 1) = 7[x(x + 1)]
Expanding:
24x + 12 = 7x^2 + 7x
Rearranging the terms to form a quadratic equation:
7x^2 + 7x - 24x - 12 = 0
7x^2 - 17x - 12 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula.
Considering the options, we will use factoring. We need to find two numbers whose product is -84 (the product of the coefficients of x^2 and the constant term -12), and whose sum is -17 (the coefficient of x). The numbers are -21 and 4:
(7x + 4)(x - 21) = 0
Setting each factor to zero, we get:
7x + 4 = 0 or x - 21 = 0
Solving each equation:
7x = -4 or x = 21
Dividing the first equation by 7:
x = -4/7 or x = 21
Since we are looking for two consecutive integers, a negative value (-4/7) doesn't make sense in this context. Therefore, the first integer, x, is equal to 21.
The second consecutive integer, x + 1, is:
21 + 1 = 22
So, the two consecutive integers are 21 and 22.