Quotient

Introduction

Division is splitting (dividend) into equal parts by a known number of parts (divisor). Division is used everywhere in real life. When a number is divided by the same number the result is 1. Example: 4\/4 = 1. When a number is divided by 1 the result is the same number. Example:15/1=15 . When 0 is divided by a number, the result is 0. Example: 0÷14 = 0. When a number is divided by 0, the result doesn't have any value. Example: 7÷0 = undefined.

Division Algorithm

To divide a number there are two steps to follow −

Step1: The long division method

The long division method is used to divide a large integer into smaller integers that cannot be divided anymore.

Step2: Division algorithm

The division algorithm states, " Let a, b be integers with b >0 then there exist unique integers q and r such that

• a = b q + r

• $\mathrm{0 \leq r < 0}$

There are four major terms used in division. They are Dividend, divisor, quotient and remainder.

In simpler form we can write division algorithm as

dividend(a) = (divisor (b) × quotient) + remainder

Terms	Definition	Denote
Dividend	The integers that need to be divided.	a
Divisor	The integer that divides the dividend.	b
Quotient	The new integer formed by dividing two numbers.	q
Remainder	The integer that cannot be divided further.	r

Quotient

The term quotient is common in division. Quotient is the integer obtained from division of two integers. When a number divides another number with zero remainder, then the Quotient is the answer. When a number divides another number with a remainder value, then the Quotient is the whole number in the mixed fraction, or it will be a decimal number. A quotient can be a larger number to divisor or integer (b), but it is a smaller number than dividend or integer (a).

Quotient as a fraction

When a number is divided by another number, the whole number results in a fraction. When a fraction is simplified, if a fraction obtained is a proper fraction, then the result is quotient. If the result is in an improper fraction, then convert it to a mixed fraction as the whole number in the mixed fraction is the quotient. To convert an improper fraction to a mixed fraction, we need a division algorithm.

Example

• $\mathrm{3÷5 = \frac{3}{5} (a\: proper\: fraction)}$

• $\mathrm{12÷4 = \frac{12}{4}=3 (a\: whole\: number)}$

• $\mathrm{43÷6=7\frac{1}{6}( a\: mixed\: fraction)}$

If a fraction is divided by another fraction to find the quotient of a fraction, there are three steps to follow −

Step 1 − Take the multiplicative inverse of the fraction by changing it to Multiplication and the reciprocal of the second number needs to be written.

Step 2 − Now, multiply the numerators and denominators.

Step 3 − Take HCF of the Numerator and denominator to find the simplest form.

The result obtained is the quotient of the fraction.

Example: Find the quotient of $\mathrm{\frac{15}{4}÷\frac{4}{5}}$

Solution

Take the multiplicative inverse,

$$\mathrm{\frac{15}{4}÷\frac{4}{5}=\frac{15}{4}\times \frac{5}{4}}$$

Multiply the numerators and denominators

$$\mathrm{\frac{15\times 5}{4\times 4}=\frac{75}{16}}$$

Simplify by taking HCF

HCF of (75,16) = 1

$$\mathrm{\frac{75\times 1}{16\times 1}=\frac{75}{16}}$$

To convert the improper fraction, use long division method

$$\mathrm{\:\:\:\:\:\:\:4\\\:16)\overline{75}\:\\\:\:\:\:\:\:\:\underline{64}\\\\\:\:\:\:\:\underline{11}}$$

by division algorithm,

$$\mathrm{\frac{dividend}{divisor} = quotient +\frac{remainder}{divisor}}$$

$$\mathrm{=4\frac{11}{16}}$$

Quotient as a decimal

Decimal is composed of a whole number part and fractional number part. The decimal point separates the whole number and fractional number.

• The numbers after decimal point have different place value starts from tenth $\mathrm{(\frac{1}{10})}$, hundredth $\mathrm{(\frac{1}{100})}$, thousandth $\mathrm{(\frac{1}{1000})}$ the value goes on from the left to right which is a fractional part.

• The numbers before the decimal point have place value starts from ones (1), tens (10), hundreds (100), thousands (1000) the value goes from right to left which is a whole part.

When two integers are divided, the answer must be a whole number, if not then the answer will be in a decimal number. When an integer is divided by another integer, if it's not having any common multiples a decimal point must be added to the quotient and 0 to the remainder value.

Example: Find the quotient 16÷3

Solution

$$\mathrm{\:\:\:\:\:\:\:5.33\\\:3)\overline{16}\:\\\:\:\:\:\:\underline{15}\\\\\:\:\:\:\:\ 10\\\\\:\:\:\:\:\:\underline{09}\\\\\:\:\:\:\:\:\ 10\\\\\:\:\:\:\:\:\:\underline{09}\\\\\:\:\:\:\:\:\:\:\ 1}$$

When 16÷ 3 the quotient is in decimal 5.33.

To find the quotient as a when a decimal number divided by another decimal number, steps are follows

• Step 1: check the divisor is a whole number, if not convert it to whole number.

• Step 2: multiply the power of 10’s to remove the decimal from the divisor.

• Step 3: multiply with the same power of 10’s to the dividend.

• Step 4: use long division

• Step 5: check the decimal point in the quotient as in the dividend.

Solved Examples

1. Find the quotient of 12.53÷3.5

Solution

To convert a divisor from decimal to whole number, multiply it by powers of 10.

3.5 ×10¹= 35

now multiply the dividend with the same powers of 10,

12.53 × 10¹= 125.3

by long division,

$$\mathrm{\:\:\:\:\:\:\:3.58 \\\:35)\overline{125.3}\:\\\:\:\:\:\:\:\:\underline{105}\:\:\:\\\\\:\:\:\:\:\:\:\:\ 203\:\\\\\:\:\:\:\:\:\:\:\:\underline{175}\\\\\:\:\:\:\:\:\:\:\:\:\ 280\\\\\:\:\:\:\:\:\:\:\:\:\:\underline{280}\\\\\:\:\:\:\:\:\:\:\:\:\:\:\underline{0}}$$

The quotient from dividing 125.3 by 35 is 3.58

Conclusion

The quotient of a number is the answer when a number is divided by another number, it will be either as a whole number or it will be a fractional number or decimal number. Division algorithm is used to convert an improper Fraction to a mixed fraction, which helps to find the whole part (quotient) in a fraction. We can obtain a quotient as a fraction as well as a decimal number.

FAQs

1. What is a remainder?

A remainder is a part of a piece or quantity which cannot be divided anymore into smaller pieces. In a mixed fraction, a remainder will be in a numerator place.

2. When will the value of a quotient will be near to 1?

When the value of numerator and denominator are same then, the value of quotient will be equal to 1. If a value of numerator and denominator are close to each other, then the value will be close to 1.

3. When a quotient will be greater than 1?

If a value of a dividend is greater than the value of Divisor, then the Quotient is greater than 1. Example: 15÷3 = 5, here 5 is greater than 1.

4. How does a quotient look when a smaller number is divided by a larger number?

The quotient will be lesser than 1 when a smaller number is divided by a larger number. Example: 10÷ 100 = 1÷10 = 0.1 which is less than 1.

5. How to find a quotient compatible?

A quotient of a number can be easily found by finding the number which is closer to divide. For example, to find the quotient of 18÷4, the near number is 16. Therefore, 16 is its compatible number.