Problem 113 - Algebra and Trigonometry

Question

Exercises 113–115 will help you prepare for the material covered in the next section. Divide 737 by 21 without using a calculator. Write the answer as quotient \(+\frac{\text { remainder }}{\text { divisor }}\)

Step-by-Step Solution

Answer

The answer is 35 + \(\frac{2}{21}\).

Step 1: Division

Begin with the long division process. Calculate how many times 21 can evenly fit into 737. This gives you a quotient of 35.

Step 2: Multiplication & Subtraction

Multiply the quotient by the divisor and subtract the result from the initial number. This provides the remainder: \((35 * 21 = 735)\), \((737 - 735 = 2)\). So, the remainder is 2.

Step 3: Quotient and Remainder forming

Put the quotient and the remainder into a fraction form with the divisor as the denominator. This results in: 35 + \(\frac{2}{21}\).

Key Concepts

Division without Calculator

Performing division without a calculator may seem daunting at first, but with the technique of long division, it becomes a systematic process that can be applied to any division problem. To begin, determine how many times the divisor can fit into the first digits of the dividend. If the divisor is larger than the first digit, consider the first two digits, and so forth.

Once you find how many times the divisor fits, write that number—the quotient—above the dividend. Next, multiply the divisor by the quotient and write the result under the dividend. Subtract this product from the initial segment of the dividend. This gives you the first part of your answer and also what to subtract from the starting number.

If there's a remainder, bring down the next digit of the dividend if available and repeat the process until you've processed all digits of the dividend. The long division method is clear-cut and, with practice, becomes an efficient tool for dividing without a calculator.

Finding Remainder

The remainder in a division problem is what's left over after dividing the dividend by the divisor. It's an important concept in mathematics because it tells us that the division isn't exact and there's a portion of the dividend that isn't wholly divisible by the divisor.

To find the remainder, first complete the process of division as far as possible, which means subtracting the largest multiple of the divisor from the dividend without exceeding it. Whatever is left after this subtraction is the remainder. In our specific exercise with dividing 737 by 21, we noticed that after subtracting 735 (21 multiplied by the quotient, 35) from 737, we are left with 2. This number 2 is then our remainder, indicating that 737 is not fully divisible by 21 and two units are left undivided.

Writing Mixed Numbers

When the division between two numbers doesn't result in a whole number, we often express the answer as a mixed number. A mixed number is a combination of a whole number and a fraction. To write the answer as a mixed number, you take the quotient from your division as the whole number part, and then the remainder becomes the numerator of the fraction, with the divisor as the denominator.

For example, from our division of 737 by 21, we determined a quotient of 35 and a remainder of 2, resulting in a mixed number of 35 and \(\frac{2}{21}\). This format is particularly useful when we wish to denote exact values, as it clearly displays the whole units and the extra fractional part.