**Category : **4th Class

Distributing a quantity into some equal parts such that each part contains equal amount is called division.

In another way, you are given 8 chocolates and told to distribute them among your four friends such that each of them gets equal number of chocolates. Then you will give 2 hocolates to each of your friend. You separate a quantity (Here 8 chocolates) into four parts (Here your friends) such that each of the parts contains equal amount,

**You have 60 books and you want to keep them into 5 drawers such that each drawer contains equal number of books. How will you find how many books should you keep in each drawer? **

**Solution:** To find solution first see the quantity that is to be distributed (Here 60 books) then find the number of parts among which quantity is to be distributed (Here 5 drawers)

Now divide the quantity by the number of parts. Therefore, the amount each \[\text{Parts contains}=\frac{\text{Total quantity}}{\text{Number of parts}}=\frac{60}{5}=12\]

By the division we easily find that we should keep 12 books in each drawer.

**Terms Related to Division**

The quantity that is to be divided is called DIVIDEND. Number of parts in which the quantity is to be divided is called DIVISOR. The amount that each group gets is termed as QUOTIENT. The extra amount which is left after equal distribution is called REMAINDER.

**Divide 8 by 3 and find the terms, dividend, divisor, quotient, remainder.**

**Solution:** \[Divisor\,\to 3\overset{2}{\overline{\left){\begin{align} & 8 \\ & \underline{6} \\ & 2 \\ \end{align}}\right.}}\begin{matrix} \to Quotient \\ \to Dividend \\ \begin{align} & \\ & \to \operatorname{Re}mainder \\ \end{align} \\ \end{matrix}\]

Here, Dividend = 8, Divisor = 3, Quotient = 2, Remainder = 2

**Relation between the Terms of Division**

\[\text{Dividend}=\text{Divisor}\times \text{Quotient}+\text{Remainder}\]

**Divide 504 by 15 and verify the relation**

**Solution:** \[\begin{align} & 15)5\,0\,4(33 \\ & \,\,\,\,\,\,\,\frac{-\,4\,5}{\begin{align} & 0\,5\,4 \\ & \frac{-\,4\,5}{\,\,\,0\,9} \\ \end{align}} \\ \end{align}\]

Dividend = 504, Divisor = 15, Quotient = 33, Remainder = 9

Thus \[\text{5}0\text{4}=\text{15}\times \text{33}+\text{9}\]

Or \[\text{5}0\text{4}=\text{495}+\text{9}\]Or\[\text{5}0\text{4}=\text{5}0\text{4}\]

**Operation on Division**

Divide 256458 by 35.

**Step 1: **Consider the first digit of the dividend from left. If the digit is smaller than the divisor, multiply the divisor by 0. Place the product below the number and the multiplier in the quotient side. Now subtract the product from the number.

\[\begin{align} & \text{35})\text{256458}(0 \\ & \,\,\,\,\,\,\,\frac{-\,\,0}{\,\,\,\,2} \\ \end{align}\]because 2 (the first digit of the dividend from left) is smaller than 35

**Step 2:** Bring down the second digit of the dividend and write it right to the difference. Now compare the formed number with divisor and if the number is smaller than the divisor, repeat the first step

\[\begin{align} & \text{35})\text{256458}(00 \\ & \,\,\,\,\,\,\,\frac{-\,\,0}{\begin{align} & \,\,\,\,2\,\,5 \\ & \underline{-\,\,\,\,0} \\ & \,\,\,2\,\,5 \\ \end{align}} \\ \end{align}\]because 25 is smaller than 35

**Step 3:** Bring down the next digit of the dividend and compare the formed number with divisor. If the number is greater than the divisor, multiply the divisor by a number such that the multiple should be closest to the number but not greater. Write the multiplier in the quotient side and the product below the number and subtract.

\[\begin{align} & \text{35})\text{256458}(00\text{7} \\ & \,\,\,\,\,\,\,\frac{-\,\,0}{\begin{align} & \,\,\,\,2\,\,5 \\ & \underline{-\,\,\,\,\,\,0} \\ & \,\,\,2\,5\,6 \\ & \underline{-245} \\ & \,\,\,\,011 \\ \end{align}} \\ \end{align}\]

**Step 4:** Bring down the next digit of the dividend and continue the process as far as all the digits of the dividend is brought down.

**Solution:**

\[\begin{align} & \text{35})\text{256458}(00\text{72} \\ & \,\,\,\,\,\,\,\frac{-\,\,0}{\begin{align} & \,\,\,\,2\,\,5 \\ & \,\,\,\underline{-\,\,\,0} \\ & \,\,\,2\,5\,6 \\ & \underline{-245} \\ & 0114 \\ & \underline{-105} \\ & 0095 \\ & \underline{-\,\,7\,\,0} \\ & \,\,\,\,258 \\ & \underline{-245} \\ & \,\,\,\,013 \\ \end{align}} \\ \end{align}\]

**Divide 68795662 by 25**

**Solution:**

\[\begin{align} & \text{25})\text{687656}(0\text{275}0\text{6} \\ & \,\,\,\,\,\,\,\frac{-\,\,0}{\begin{align} & \,\,\,6\,8 \\ & \underline{-5\,0} \\ & \,\,\,\,\,187 \\ & \underline{-175} \\ & 0126 \\ & \underline{-125} \\ & 0015 \\ & \underline{-\,\,\,\,\,0} \\ & \,\,\,\,156 \\ & \underline{-150} \\ & \,\,\,\,006 \\ \end{align}} \\ \end{align}\]

**Word Problem Based on Division**

**There are 1314936 apples to be kept in the boxes. If one of the box can contain 24 apples, find the number of required boxes to keep all the apples.**

**Solution:**

One box can contain 24 apples

Total number of apples = 1314936

Therefore, required number of boxes\[=\frac{1314936}{24}=54789\]

**Each of the student in a class contributed Rs. 22 for a tour. Thus total Rs. 9081248 was collected. Find the number of students in the class. **

**Solution:**

Total moneys Rs. 9081248

Money contributed by each of the students = Rs. 22

Thus, number of students\[=\frac{9081248}{22}\]

**Word Problems Based on Multiplication and Division**

**There are 4562 people in a small town. Each people contributes Rs. 50 to plant trees around the town. If planting one tree costs Rs. 25 find the maximum number of trees which can be planted by the total collected money.**

**Solution:** Total collected money =Rs. \[\text{4562}\times \text{5}0\] = Rs. 228100

\[\because \]Planting one tree costs Rs. 25

\[\therefore \]Number of trees which can be planted by Rs. 228100

\[=\frac{228100}{25}=9124\]

**A book of mathematics has 175 pages. Each of the pages contains 28 questions. Cody solves 25 questions each day. How many days will he take to solve all the questions?**

**Solution:** Number of questions in 1 page =28

Total number of pages = 175

Thus, total number of questions \[=\text{175}\times \text{28=4900}\]

\[\because \]Cody solves 25 questions each day

\[\therefore \]Required number of days\[=\frac{4900}{25}=196\]

- Multiplication means repeated addition.
- In multiplication, the number which is multiplied is known as MULTIPLICAND.
- The numbers by which it is multiplied is known as MULTIPLIER and the answer or the result of multiplication is known as PRODUCT.
- Division can be called repeated subtraction.
- The number which is divided is called dividend.
- The numbers by which the dividend is divided is called divisor.
- The number by which the divisor is multiplied to get closest to the dividend.
- The number which left after subtraction is called reminder is called quotient.

- The early Indian mathematician to develop a variety of intuitive tricks to perform multiplication. They use slate hand and small tablets to perform calculations.
- The method of multiplication which we use now a day was first described by the Indian mathematician Brahmgupta.

**6 + 6 + 6 + ....... 695 times = ? **

(a) 4170

(b) 4200

(c) 4270

(d) 4110

(e) None of these

**Answer (a)**

**Explanations**

\[\text{6}+\text{6}+\text{6}+...\text{ 695}\]times\[=\text{6}\times \text{695}\] = 4170.

**In a book shop there are 43 books each with 31 pages and 87 other books each with 41 pages, what is the total number of pages of the books in the bookshop. **

(a) 4900

(b) 4564

(c) 4954

(d) 4520

(e) None of these

**Answer (a)**

**Expiation**

Total number of pages \[=\text{31}\times \text{43}+\]\[~\text{41}\times \text{87}=\text{1333}+\text{35671}=\text{49}00\]

**A long distance train has 93 bogies and each bogey has 32 passengers. Find the number of passengers in the train?**

(a) 2954

(b) 2986

(c) 3001

(d) 2976

(e) None of these

**Answers (d)**

**Explanations**

Passengers in 1 bogey = 32 Passengers in 93 bogies\[=\text{93}\times \text{32}=\text{2976}\]

**Martin goes to market every fifth day and spends Rs. 500 on purchasing. How much money in a year does he spend on purchasing? **

(a) Rs. 36500

(b) Rs. 35500

(c) Rs. 3600

(d) Rs. 36000

(e) None of these

**Answer (a)**

**Which digit should come in place of\[\], so that following multiplication is correct?**

\[\begin{align} & 4\,\,8 \\ & \underline{\,\,\,\,\times \,\,\,8} \\ & \underline{3984} \\ \end{align}\]

(a) 1

(b) 5

(c) 7

(d) 9

(e) None of these

**Answer (d)**

**Find the remainder when 1461984 is divided by 32.**

(a) 24

(b) 0

(c) 13

(d) 18

(e) None of these

**Answer (b)**

**Explanation -**

\[=\frac{1461984}{32}=45687\]

We find 32 completely divides 1461984 and leaves no remainder.

**When Cody multiplies A by 59 the product he gets 322376. Find the value of A.**

(a) 5406

(b) 4123

(c) 3214

(d) 5464

(e) None of these

**Answer: (d)**

**Explanation -**

According to the question \[\text{A}\times \text{59}=\text{322376}\],\[~\text{A}=\frac{\text{322376}}{59}=\text{5464}\] Therefore, option (d) is correct and rest of the options is incorrect.

** Find the remainder when 1461982 is divided by 32.**

(a) 29

(b) 30

(c) 31

(d) 32

(e) None of these

**Answer: (b)**

**There are 45680 bags of cement. Each bag contains 60 kg cement. Jack opens all the bags and put the cement into 50 containers equally. Find the amount of cement each container has.**

(a) 54816kg

(b) 45623kg

(c) 65231kg

(d) 57623kg

(e) None of these

**Answer: (a)**

**A car can travels the distance 75 km in 1 hour. To cover the distance 681525 km, how much time will the car take?**

(a) 4568 hours

(b) 9087 hours

(c) 7896 hours

(d) 4623 hours

(e) None of these

**Answer: (b)**

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