Problem 570 - Prealgebra

Question

Divide. Then check by multiplying. $$\frac{355}{5}$$

Step-by-Step Solution

Answer

The quotient is 71. Checking by multiplication, 71 multiplied by 5 equals 355.

Step 1: Divide

First, divide the numerator (355) by the denominator (5). Perform the division: \[ 355 \text{ divided by } 5 = 71 \] So, the quotient is 71.

Step 2: Multiply to Check

Multiply the quotient (71) by the original denominator (5) to check if you get the original numerator (355): \[ 71 \times 5 = 355 \] Since the result is 355, the division is correct.

Key Concepts

Division

Division is the process of finding out how many times one number is contained within another. It is one of the basic arithmetic operations, alongside addition, subtraction, and multiplication. When we divide, the number being divided is called the 'numerator' and the number we are dividing by is called the 'denominator'. Using our example, we divide 355 by 5. The process can be shown like this:

• First, see how many times 5 goes into 35, which is 7 times.
• Then, bring down the next digit (5) to see how many times 5 fits into this number.
• It fits exactly 1 time.

So, we put these results together to find that \[ 355 \div 5 = 71 \] The answer is 71. Division helps us break down problems and distributes a number into equal parts.

Multiplication

Multiplication is another basic arithmetic operation, which is essentially repeated addition. If you multiply 3 by 4, it means that you add 3 four times (e.g., 3 + 3 + 3 + 3). To verify our division, we use multiplication. We take the quotient we found (71) and multiply it by the original denominator (5) to ensure it equals the numerator (355). The step looks like this: \[ 71 \times 5 = 355 \] This confirms our division because the result is the same as our original numerator. Thus, multiplication helps verify our division since it reverses the process. The correct product assures that division was done correctly and adds a layer of assurance to the solution.

Numerator and Denominator

In mathematics, especially in fractions and ratios, the terms 'numerator' and 'denominator' are crucial. The numerator is the number above the fraction line, representing the part of the whole we are considering. The denominator is the number below the fraction line, representing the total number of equal parts. In our example, \[ \frac{355}{5} \] - 355 is the numerator (the number being divided) - 5 is the denominator (the number we are dividing by) Understanding these terms helps in executing operations like division correctly. Knowing which number is which is vital for proper calculation and understanding the relationship between the parts of our fractions.