The sum of a number and its inverse is $3 \frac{29}{52}$.
Answer (2)
Convert the mixed number to an improper fraction: 3 52 29 = 52 185 .
Set up the equation: x + x 1 = 52 185 .
Transform the equation into a quadratic equation: 52 x 2 − 185 x + 52 = 0 .
Solve the quadratic equation using the quadratic formula to find the solutions: x = 4 13 or x = 13 4 .
The number is: 4 13 or 13 4 .
Explanation
Problem Setup Let x be the number. The problem states that the sum of the number and its inverse is 3 52 29 . We can write this as an equation: x + x 1 = 3 52 29
Converting to Improper Fraction First, convert the mixed number to an improper fraction: 3 52 29 = 52 3 × 52 + 29 = 52 156 + 29 = 52 185 So the equation becomes: x + x 1 = 52 185
Clearing Fractions and Rearranging To solve for x , multiply both sides of the equation by 52 x to eliminate the fraction: 52 x ( x + x 1 ) = 52 x ( 52 185 ) 52 x 2 + 52 = 185 x Rearrange the equation to form a quadratic equation: 52 x 2 − 185 x + 52 = 0
Applying the Quadratic Formula Now, solve the quadratic equation 52 x 2 − 185 x + 52 = 0 using the quadratic formula: x = 2 a − b ± b 2 − 4 a c where a = 52 , b = − 185 , and c = 52 .
Calculating the Discriminant Calculate the discriminant: D = b 2 − 4 a c = ( − 185 ) 2 − 4 ( 52 ) ( 52 ) = 34225 − 10816 = 23409 Since 23409 = 153 , we have: x = 2 ( 52 ) 185 ± 153 = 104 185 ± 153
Finding the Solutions Find the two possible values of x : x 1 = 104 185 + 153 = 104 338 = 4 13 x 2 = 104 185 − 153 = 104 32 = 13 4
Final Answer The two solutions are x 1 = 4 13 and x 2 = 13 4 . These are reciprocals of each other, so either value is a valid solution.
Conclusion Therefore, the number is either 4 13 or 13 4 .
Examples
Understanding the relationship between a number and its inverse is useful in various fields. For example, in electrical engineering, the reciprocal of resistance is conductance. If you know the total resistance in a circuit and need to find the conductance, you would use the concept of inverse. Similarly, in finance, understanding reciprocals can help in calculating ratios and proportions, such as comparing earnings per share to price-to-earnings ratios. This concept also applies in physics when dealing with quantities like frequency and period, which are reciprocals of each other.
The number is either 4 13 or 13 4 , since both solutions satisfy the equation formed by the sum of the number and its inverse equaling 3 52 29 .
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