Problem 102 - Elementary Algebra

Question

Perform the operations. Reduce answers to lowest terms. $$ -59 \div 53 \cdot 52 $$

Step-by-Step Solution

Answer

The final result is -58.

Step 1: Perform Division

Begin by performing the division operation given in the expression. For the expression \[-59 \div 53,\]rewrite it as a fraction:\[\frac{-59}{53}.\] Since it cannot be simplified further, we'll leave it as is.

Step 2: Multiply the Fraction

Now, multiply the result from Step 1 by the remaining number:\[\frac{-59}{53} \cdot 52.\]Treat 52 as a fraction \(\frac{52}{1}\) and perform the multiplication:\[\frac{-59 \times 52}{53 \times 1} = \frac{-3068}{53}.\]

Step 3: Simplify the Expression

Next, check if the resulting fraction \(\frac{-3068}{53}\) can be simplified. Since 53 is a prime number, any common factor would have to be present in 3068 as well. By dividing 3068 by 53, we find:\[3068 \div 53 = 58,\]so the expression simplifies to:\[-58.\]

Key Concepts

Division of Integers

When dividing integers, the process involves splitting one whole number by another. In our example, \(-59 \div 53\), we interpret this as dividing -59 by 53. To express this division, write the operation as a fraction: \(\frac{-59}{53}\).
A fraction essentially represents division. In this case, the numerator -59 is divided by the denominator 53. Since 59 and 53 have no common factors other than 1, this fraction is in its simplest form already, which means it cannot be reduced further. Even though fractions can look complex, if the numerator and the denominator share no common factors (except for 1), the fraction is considered simplified.

Multiplication of Fractions

Multiplying fractions might seem a bit daunting, but it follows a straightforward method. Imagine we begin with two fractions \(\frac{-59}{53}\) and treat 52 as \(\frac{52}{1}\), since any whole number can be expressed as a fraction with 1 in the denominator.
To multiply these fractions, simply multiply the numerators together, and multiply the denominators together separately: \[-59 \times 52 = -3068\] and \[53 \times 1 = 53\].
This gives you the new fraction \(\frac{-3068}{53}\). Remember, when multiplying fractions, the result is simply the product of the numerators over the product of the denominators.

Simplifying Fractions

Simplifying fractions is crucial to arrive at the most reduced form, making calculations and understanding easier. Our end result from the multiplication step is the fraction \(\frac{-3068}{53}\). To simplify, we need to determine if the numerator 3068 can be evenly divided by the denominator 53.
Since 53 is a prime number, any factors 3068 shares with 53 can help us simplify. By performing the division, \(3068 \div 53\), we find the result is 58.
This division indicates 3068 is exactly 53 multiplied by 58, so they share the common factor 53. Simplifying \(\frac{-3068}{53}\) gives us the integer -58.
When simplifying fractions, always check if the numerator is a multiple of the denominator. If it is, division yields a simpler form, often a whole number, which represents the same value more concisely.