Find the sum and write in lowest terms.
$-\frac{23}{8}+\frac{9}{16}$
To find the sum of \(-\frac{23}{8}\) and \(\frac{9}{16}\), we must first express both fractions with a common denominator. The denominators of the two fractions are 8 and 16. The least common multiple (LCM) of 8 and 16 is 16, as 16 is a multiple of 8 (since \(8 \times 2 = 16\)).
We now convert the fraction \(-\frac{23}{8}\) to an equivalent fraction with a denominator of 16. To do this, we multiply both the numerator and the denominator by 2:
\[-\frac{23}{8} = -\frac{23 \times 2}{8 \times 2} = -\frac{46}{16}.\]
Now, both fractions have the same denominator, 16.
With a common denominator, we can now add the two fractions by summing their numerators and keeping the common denominator:
\[-\frac{46}{16} + \frac{9}{16} = \frac{-46 + 9}{16}.\]
Calculating the sum of the numerators gives \(-46 + 9 = -37\). Thus, the sum is \(\frac{-37}{16}\).
Finally, we check if the fraction \(\frac{-37}{16}\) can be simplified further. The number 37 is a prime number and is not a factor of 16 (since 16 is \(2^4\)). Therefore, the greatest common divisor of 37 and 16 is 1, which means the fraction is already in its simplest form. The result of the addition is \(\boxed{-\frac{37}{16}}\).
In real-life scenarios, adding fractions is essential for tasks like adjusting recipes or managing construction measurements. For instance, if you have a piece of lumber that is -23/8 units long (representing a deficit or a coordinate) and you add a piece that is 9/16 units long, finding the combined total helps in precise project planning. This skill ensures that your calculations remain accurate whether you are balancing a budget, measuring materials, or working with scientific data.