Login
OR
Don't worry. We'll fix this for you!
Division is one of the four fundamental operations of arithmetic. At its core, division is about splitting a number into equal parts. Imagine you have INR 100 and want to share it equally among 4 friends. How much money does each friend get? This sharing process is a simple example of division.
In division, the number you want to split is called the dividend. The number of equal parts you want to divide the dividend into is called the divisor. The result you get after dividing is called the quotient. Sometimes, after division, there's a leftover part that can't be divided equally; this leftover is known as the remainder.
Formally, division is defined as the process of determining how many times one number (the divisor) is contained within another number (the dividend).
Division Relationship: If we divide a number \(A\) (dividend) by another number \(B\) (divisor), the result is a quotient \(Q\), possibly with a remainder \(R\).
This is expressed as:
\( A = B \times Q + R \) where \(0 \leq R < B\)
Example: Suppose you have 17 apples and want to pack them equally in boxes that hold 5 apples each.
Dividing 17 by 5:
\[ 17 = 5 \times 3 + 2 \]So, you can fill 3 full boxes and have 2 apples remaining.
When dividing large numbers, the long division method helps break down the division into manageable steps. It involves dividing, multiplying, subtracting, and bringing down digits step by step until the entire dividend has been processed.
graph TD A[Start with dividend and divisor] A --> B[Divide leftmost digits by divisor] B --> C[Write quotient digit above dividend] C --> D[Multiply quotient digit by divisor] D --> E[Subtract product from dividend part] E --> F{Remaining digits left?} F -->|Yes| G[Bring down next digit] G --> B F -->|No| H[Write remainder] H --> I[End]This flow clarifies that each step relies on the previous, making the process systematic and organized.
Sometimes, either the dividend or the divisor or both may be decimal numbers (numbers with digits after the decimal point). Before division, it's often easier to convert the divisor into a whole number by shifting the decimal point to the right. To keep the ratio the same, you do the same for the dividend.
For example, dividing 56.4 by 1.2 can be simplified by multiplying both by 10 (shifting decimal one place to the right):
\[56.4 \div 1.2 = \frac{564}{12}\]Now, divide 564 by 12 using long division.
Division and multiplication are inverse operations. For any numbers \(A\), \(B\), and \(Q\):
\[A \div B = Q \quad \iff \quad B \times Q = A\]This means dividing \(A\) by \(B\) gives a number \(Q\) such that multiplying \(Q\) back by \(B\) returns \(A\).
When dividing fractions, things become easier by using the reciprocal of the divisor.
The reciprocal of a fraction \(\frac{c}{d}\) is \(\frac{d}{c}\). Dividing a fraction by another fraction is the same as multiplying by the reciprocal of the divisor:
\[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\]This operation often confuses students, but remembering "divide by a fraction, multiply by its reciprocal" quickly resolves it.
| Operation | Expression | Equivalent Multiplication |
|---|---|---|
| Dividing Whole Numbers | 20 / 4 | Find \(Q\) such that \(4 \times Q = 20\) |
| Dividing Fractions | \(\frac{3}{4} \div \frac{2}{5}\) | \(\frac{3}{4} \times \frac{5}{2}\) |
| Dividing Decimals | 56.4 / 1.2 | Convert to \(564 \div 12\), then divide |
Step 1: Consider how many times 12 fits into the first digits. Look at '34' (first two digits of 3456). 12 goes into 34 two times (since 12 x 2 = 24), less than 3 times (which would be 36).
Step 2: Write 2 as the first digit of the quotient. Multiply 2 x 12 = 24 and subtract: 34 - 24 = 10.
Step 3: Bring down the next digit from the dividend (5), making the new number 105.
Step 4: Find how many times 12 goes into 105. It goes 8 times (12 x 8 = 96). Write 8 in the quotient.
Step 5: Subtract 105 - 96 = 9. Bring down the last digit 6, making 96.
Step 6: 12 goes into 96 exactly 8 times (12 x 8 = 96). Write 8 in the quotient.
Step 7: Subtract 96 - 96 = 0. Nothing remains.
Answer: The quotient is 288.
Step 1: Shift the decimal point in both dividend and divisor one place to the right to make divisor a whole number:
\[ 56.4 \div 1.2 = 564 \div 12 \]Step 2: Use long division to divide 564 by 12.
12 into 56 goes 4 times (12 x 4 = 48), subtract: 56 - 48 = 8. Bring down 4, making 84.
12 into 84 goes 7 times exactly (12 x 7 = 84), subtract: 84 - 84 = 0.
Answer: Quotient is 47.0 or 47.
Step 1: Here, Dividend = 4680 INR, Divisor = 15 people.
Step 2: Divide 4680 by 15 using long division or estimating:
15 x 300 = 4500; remaining 180.
15 x 12 = 180; total 300 + 12 = 312.
Answer: Each person receives INR 312.
Step 1: Divide 123 by 7.
7 x 17 = 119, remainder \(123 - 119 = 4\).
Step 2: Quotient = 17, Remainder = 4.
Interpretation: Each person receives 17 units, and 4 units remain undistributed.
Step 1: Find reciprocal of divisor \(\frac{2}{5}\), which is \(\frac{5}{2}\).
Step 2: Multiply dividend \(\frac{3}{4}\) by reciprocal \(\frac{5}{2}\):
\[ \frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8} \]Answer: \(\frac{15}{8}\) or \(1 \frac{7}{8}\) as a mixed number.
When to use: When dealing with large numbers or decimals to reduce complexity.
When to use: To avoid confusion and errors with decimal placements.
When to use: For faster mental calculation or checking answers in exams.
When to use: While solving fraction division problems to avoid mistakes.
When to use: When providing final answers especially in word problems.
Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.
Go to practice →
Your Score
Your Rank
| Rank | Name | Score |
|---|