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## Decimal is based on fractions. The number less than one is a decimal. It is denoted by a dot. Suppose the fraction is 4/5. When we divide 4 by 5, that is numerator by its denominator the answer is 0.8. This 0.8 is also 8 tenths and a decimal .

UNDERSTANDING DECIMALS

REPRESENTING DECIMAL IN TENTHS

When 3 is divided by 10, that is the numerator by its denominator it is 0.3. As 10 is greater than 3, we have to put dot to divide 3 by 10, and then we write 0 after 3, that is 30, so it is 0.30. Three zeros can be written after putting dot

REPRESENTING DECIMAL IN
HUNDREDTHS

8 divided by 100 is 0.08. First we have to put one dot, so it would be 80, but still 80 is less than 100. So we have to write zero after dot, and then we can put one more zero after 80, hence now it is 800 and the quotient is 0.08.

When 44 is divided by 100, put one zero in 44 after putting dot, so 440 divided by 100 is 0.4, and the remainder is 40, after that put zero in 40 that is 400, and then 400 divided by 100 is 4 that we have to write after 0.4, hence answer is 0.44.

Decimal can be written along with whole numbers also. 1.78 and 3.75 are such examples. Other examples are 234.67 and  346.45.

Since 0.4 covers more portion than 0.04, 0.4 is greater than 0.04

### WHICH IS SMALLER IN 2.25 AND 2.3 ADDITION OF DECIMALS  SUBTRACTION IN DECIMALS  MULTIPLICATION OF DECIMALS

Multiply 12.5 by 2
12.5
X 2
25.0

Multiplication is repeated addition alternatively when we add 12.5 two times, that is 12.5 + 12.5 = 25. When you have to multiply, multiply simply as you do, 125 X 2 is 250, put a dot after one digit from the right side in answer that is 250, as there is one digit after decimal in 12.5

If you have to multiply 24.56 by 3 then
24.56
X  3
73.68
There are two digits after decimal point that is 5 and 6, hence we have to put decimal after two digits from the right side, or 24.56 + 24.56 + 24.56 = 73.68

Multiply 35.6 by 5, 124.8 by 9, multiply 29.6 by 8

Multiply 256.78 by 42.90. Multiplying 256.78 by 42.90 is same as multiplying 256.78 by 42.9. 42.90 = 42.9.

256.78
X 42.9
11015.862
There is two digits after dot in 256.78 and one digit after dot in 42.9 So total there are three digits after dot, hence multiply normally and then put dot after three digits from the right side in the answer .

345.67
X 213.45
73783.2615
Multiply as you do and total there are four digits after decimal point, two in 345.67 and two in 213.45, so put decimal point after four digits from the right side, in the answer .

TYPES OF DECIMALS

(A) TERMINATING DECIMALS

These decimals are the decimals, that have limited number of decimal points, or the decimal points end after certain number when numerator is divided by its numerator. Suppose the fraction is 6/8, so when 6 is divided by 8, it is 0.75, that is the decimal ends here.
Taking example of mixed fraction suppose 128/50, so 128 divided by 50 is 2.56, or 569/16 = 35.6525.

So the decimal points ends after certain number and the number of digits after decimal points can be counted, as in 35.6525 the number of digits after decimal points is 4 that is 6, 5, 2 and 5. These are also known as exact decimal, as numerator is divided by its denominator exactly.

(B) NON - TERMINATING DECIMALS

These are non terminating decimal numbers as the digits after the decimal point never end or terminate . The numerator is not divisible by its denominator exactly. There are countless number of decimal points after the decimal point. For instance the numerator 125 is divided by 9 it is 13.888888, and it goes on for countless number of times . These decimals are also of two types .

1. RECURRING DECIMAL NUMBERS
The same single digit or set of digits repeat or re occur after the decimal points, hence it is recurring decimal numbers. As an example when 625 is divided by 15, it is 41.666666666, that is 6 the same single digit is repeating itself infinite or countless number of times, and an order of repeating the digit 6 is followed

If 90 is divided by 11, it is 8.181818181818 the set of numbers that is 8 and 1 is repeated endless number of times and a specific order or pattern is followed .

2. NON RECURRING DECIMAL NUMBERS
In this type of decimal numbers, same digits are not repeated, nor any specific order or pattern is followed. The number of digits are countless after the decimal point. For instance 321/46 is 6.97826086956217. Any order is not followed in this .

CONVERSION OF DECIMALS
TO FRACTIONS

Converting decimal to fractions is easy and simple

1. CONVERT 0.2 AND 3.2 TO A FRACTION
Converting 0.2 to a fraction, observe how may digits are after the decimal point. It is one digit, that is 2. Write 0.2 as 2. This 2 becomes the numerator of the fraction. Write 1 and since there is one digit after the decimal point put one zero after 1 hence it is 10, that is the denominator of the fraction. So the fraction is 2/10. We can further reduce the fraction to lowest term divide 2 and 10 by the same number that is 2, hence it is 1/5 .

If 3.2 is to be changed into a fraction, same process it to be followed, hence the fraction is 32/10 and then dividing 32 and 10 both by 2, it becomes 16/5 .

2. CONVERT 0.75 AND 4.75 TO A FRACTION
When we have to convert 0.75 to a fraction we have to see how many digits are after decimal point. There are two digits that is 7 and 5 after the decimal point. Write 0.75 as 75 which is the numerator of the fraction. So just write 1 and since there are two digits after decimal point, write two zeros after 1, so it is 100 that is the denominator of the fraction. Therefore fraction is 75/100. Reducing the fraction to lowest term is dividing 75 and 100 by the same number that is 25 is 3/4.

Same process to be followed for converting 4.75 to a fraction, so it becomes 475/100, dividing 475 and 100 by same number that is 25, reduced to lowest term is 19/4

3. CONVERT 0.875 AND 52.875 TO A FRACTION
There are three digits after decimal point in 0.875. Write 0.875 as 875, and put three digits after 1, which is 1000. So the fraction becomes 875/1000. Dividing 875 and 1000 by the same number that is 125, so reducing to lowest term it is 7/8

Converting 52.875 to fraction is 52875/1000, and reducing it to lowest term is 423/8.

QUESTIONS BASED ON DECIMALS

Q – 1. Convert 425/4 into a decimal

(a) 106.20

(b) 116.25

(c) 216.35

(d) 106.25

(a) 40

(b) 45

(c) 32

(d) 27

Q – 3. How much grams is 2.25 of 3 kilograms ?

(a) 5,760 grams

(b) 6,800 grams

(c) 6,750 grams

(d) 4,500 grams

Q – 4. If you shade 0.65 units in this rectangle, then how many units would not be shaded ?

(a) 63

(b) 60

(c) 117

(d) 108

Q – 5. Farhaan has Rs 250, and Joya has double the amount that Farhaan has . Both Farhaan and Joya spends 0.6 and 0.8 respectively of the money they have. Which of the following statement is correct according to the information given ?

(a) Joya has Rs 50 more than Farhaan has left after spending

(b) Farhaan has Rs 20 less than Joya has left after spending

(c) Both of them have same amount left after spending

(d) Joya has Rs 100 more than Farhaan has left after spending

Q – 6. Ramesh has Rs 6,000 . He spends 0.25 of it on education, 0.50 of its remaining on purchase of groceries. From the amount left he spends 3/5th on shopping of clothes and rest he deposits in bank. How much amount he deposits in bank ?

(a) Rs 1,000

(b) Rs 900

(c) Rs 1,500

(d) Rs 500

Q – 7. How many lesser units must be shaded so that 0.6 of the units are shaded in this figure ?

(a) 8

(b) 12

(c) 10

(d) 15

Q – 8. How much is 2.06 of 360 less than 2.6 of the same number ?

(a)  194.4

(b) 195.4

(c) 294.04

(d) 194.04

Q – 9. Rahesh has 40 pieces of chocolate. Out of that  he distributes 0.75 of the pieces and eats 0.5 of the pieces left on Friday. How many pieces Rahesh eats on Friday ?

(a) 3

(b) 4

(c) 5

(d) 6

Q – 10. The price of 3.8 kilograms mangoes ten days ago was Rs 152. Now at present the price of mangoes has increased by 0.30 How much is the present price of 1.2 kg mangoes ?

(a) Rs 60.5

(b) Rs 52.4

(c) Rs 62.4

(d) Rs 56.8

ANSWERS WITH EXPLANATIONS

Q - 1. (d) When 425 is divided by 4 it is 106.25

Q - 2. (b) There are total 25 diamonds, its 1.8 is 1.8 x 25 = 45

Q - 3. (c) 3 kilograms are 3 X 1,000 = 3,000 grams. Its 2.25 is 2.25 X 3,000 = 6,750 grams

Q - 4. (a) Total there are 180 units, as it is grid of 18 by 10. Units shaded is 0.65, hence units not shaded is 1 - 0.65 = 0.35. Hence number of units not shaded is 0.35 X 180 = 63 units

Q - 5.(c) Farhaan has Rs 250. Amount Joya has is double of the amount that Farhaan has, so amount Joya has 250 X 2 = Rs 500. Farhaan spends 0.6, so amount he spends is 250 X 0.6 = Rs 150. Hence amount Farhaan has left is Rs 250 - Rs 150 = Rs 100. Amount Joya spends is 0.8, so she spends 0.8 X 500 = Rs 400. Thus amount Joya has left is Rs 500 - Rs 400 = Rs 100. So both of them have same amount left after spending that is Rs 100.

Q - 6.(b) Amount Ramesh has Rs 6,000. Amount spend on education is 0.25, which is 0.25 X 6000 = Rs 1,500. So amount remaining is 6,000 - 1,500 = Rs 4,500. Amount spent on purchase of groceries is 4,500 X 0.5 = Rs 2,250. Thus amount left is Rs 4,500 - Rs 2,250 = Rs 2,250. Hence amount spent on shopping of clothes is 3/5 X 2,250 = Rs 1,350. Therefore amount Ramesh deposits in bank is Rs 2,250 - Rs 1,350 = Rs 900.

Q - 7.(b) Total units are 480 as it is a grid of 32 by 15. To shade 0.6 units, units shaded must be 0.6 X 480 = 288. Units shaded is 300, hence number of units shaded lesser must be 300 - 288 = 12

Q - 8.(a)  2.06 of 360 is 2.06 X 360 = 741.6.  2.60 of 3.6 is 2.6 X 3.6 = 936. So 2.06 of 360 is less than 2.60 of 360 by 936 - 741.6 = 194.4

Q - 9.(c)  Pieces of chocolate that Rahesh distributes is 0.75 X 40 = 30. So pieces left is 40 - 30 = 10. So pieces Rahesh eats on Friday is 0.5 X 10 = 5

Q - 10.(c)  The price of 3.8 kilograms mangoes ten days ago was Rs 152, hence price of one kilogram mangoes was 152/3.8 = Rs 40. Price at present has increased by 0.30, hence present price is 40 X 1.30 = Rs 52. So present price of 1.2 kilogram mangoes is 1.2 X 52 = Rs 62.4

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