Decimals and its Operations

**Category : **7th Class

Decimal number is an another way to write a fraction. For example, $ 4.25, $ 0.25, the number after the decimal or right side from the decimal is part of one dollar. It can also be written as \[\frac{25}{100}\] of a dollar. The base of decimal number system is 10 and Indo - Arabic number system is base 10 number system. Base 10 or decimal number system can always change the place value of the number by one spot, either multiplying or dividing by 10. It is more clear from the example given below:

** Decimal point**

The decimal point is the most important part of a decimal number. It is exactly to the right of the units position.

**Like and Unlike Decimals**

The decimals having the same number of digits to the right of decimal point are called like decimals, otherwise decimals are unlike. For example 10.52, 0.63, 258.69, 35.74 are like-decimals whereas 11.205, 4.23, 7.852, 14.00087 are unlike decimals.

**Terminating and Non-Terminating and Repeating Decimals**

If the decimal representation of a fraction \[\frac{p}{q}\] comes to an end then the decimal we obtain, is called terminating decimals. Important note: A fraction \[\frac{p}{q}\] is a terminating decimal, if prime factors of q are 2 and 5 only.

**\[2\frac{3}{5}=2.6\] is a terminating decimal.**

**Non - terminating and repeating decimals:** A decimal in which digit or a set of digits repeats periodically is called non - terminating and repeating decimals. For example, \[\frac{1}{3}=0.3333333.....=\text{ }0.\overline{3}\]and\[\frac{3}{11}=0.272727.......=0.\overline{27}\] are non-terminating and repeating decimals.

**Find the terminating decimals from the following fractions: **

\[\frac{23}{25},\frac{219}{175},\frac{337}{80},\frac{29}{198},\frac{19}{512}\]

(a) \[\frac{23}{25},\frac{219}{175},\frac{337}{80}\]

(b) \[\frac{337}{80},\frac{29}{198},\frac{19}{512}\]

(c) \[\frac{219}{175},\frac{337}{80},\frac{29}{198}\]

(d)\[\frac{23}{25},\frac{337}{80},\frac{19}{512}\]

(e) None of these

**Answer:** (d)

**Explanation**

The denominators of fractions are 25,175, 80,198, 512 and their prime factors are: \[25=5\times 5\]

\[175=5\times 5\times 7,\]\[80=2\times 2\times 2\times 2\times 5,\]\[198=2\times 3\times 3\times 11\]

\[512\text{ }=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\]

Here we observe that the prime factors of 25, 80 and 512 have only 2 and 5.

Hence\[\frac{23}{25},\frac{337}{80},\frac{19}{512}\] are terminating decimal.

** Express one second into hour.**

(a) .0025 hours

(b) 1.0256 hours

(c) .00027 hours

(d) 1.000126 hours

(e) None of these

**Answer: **(c)

**Explanation**

One second \[=\frac{1}{60\times 60}=.00027\] hours

** Which one of the following fractions is in ascending order?**

(a) \[\frac{16}{19},\frac{11}{14},\frac{17}{22}\]

(b) \[\frac{11}{14},\frac{16}{19},\frac{17}{22}\]

(c) \[\frac{17}{22},\frac{11}{14},\frac{16}{19}\]

(d) \[\frac{16}{19},\frac{17}{22},\frac{11}{14}\]

(e) None of these

**Answer:** (c)

**Explanation**

\[\frac{16}{19}=0.842,\frac{17}{22}=0.773\And \frac{11}{14}=0.786.\]

\[\therefore 0.77<0.786<0.842\frac{17}{22}<\frac{11}{14}<\frac{16}{19}\]

**The LC.M. of 3, 0.09 and 2.7 is:**

(a) 2.7

(b) 1.27

(c) .027

(d) 0.27

(e) None of these

**Answer:** (d)

** The H.C.F of 0.54, 1.8 & 7.2 is:**

(a) 1.8

(b) 0.18

(c) .018

(d) 18

(e) None of these

**Answer:** (b)

**Addition and Subtraction of Decimals **

The following steps are used to add or subtract decimals

**Step 1:** Convert the decimals into like decimals

**Step 2:** Arrange them into columns in such a way that decimal point directly below each other

**Step 3**: Add or subtract as a whole number

**Step 4:** Place decimal point directly below the other decimal point in the answer

** If \[\sqrt{4096}=64\]then the value of \[\sqrt{40.96}+\sqrt{0.4096}+\sqrt{0.00004096}\]is?**

(a) 7.09

(b) 17.1014

(c) 7.1104

(d) 7.12

(e) None of these

**Answer:** (c)

**Explanation**

\[\sqrt{\frac{4096}{100}}+\sqrt{\frac{4096}{10000}}+\sqrt{\frac{4096}{1000000}}+\sqrt{\frac{4096}{100000000}}\]

\[\sqrt{\frac{4096}{10}}+\sqrt{\frac{4096}{100}}+\sqrt{\frac{4096}{1000}}+\sqrt{\frac{4096}{10000}}\]

\[=\frac{64}{10}+\frac{64}{100}+\frac{64}{1000}+\frac{64}{10000}\]

\[=6.4+.64+0.64+00.64=7.1104\]

**The following steps are used to multiply decimals: **

**Step 1:** Multiply two decimals as a whole number without bothering about the decimal point.

**Step 2:** Insert decimal point in the result by counting as many place from the right to left as the sum of the number of decimal places of the given decimals.

**Step 3:** In order to multiply decimal by 10 or its higher power, the decimal point shifted to right or left according to power is positive or negative respectively, e. g \[1.2035\times {{10}^{3}}=1203.5\]and \[1.2035\times {{10}^{-3}}=0.0012035\]

**Division of Decimals **

The following steps are used to divide a decimal by decimal.

**Step 1:** Multiply the dividend and divisor by 10 or its suitable power to convert the devisor into a whole number

**Step 2:** Divide the new dividend by the whole number obtained in step 1.

** If is \[\sqrt{5}=2.24\]then the value of \[\frac{3\sqrt{5}}{2\sqrt{5}-0.48}is:\]**

(a) 0.168

(b) 1.68

(c) 16.8

(d) 168

(e) None of these

**Answer:** (b)

**Explanation**

\[\frac{3\sqrt{5}}{2\sqrt{5}-0.48}=\frac{3\times 2.24}{2\times 2.24-0.48}=\frac{6.72}{4}=1.68\]

**If \[\sqrt{15}=3.88\] then the value of \[\sqrt{\frac{5}{3}}\] is: **

(a) 0.43

(b) 1.89

(c) 1.29

(d) 1.63

(e) None of these

**Answer:** (c)

**Explanation **

\[\sqrt{\frac{5}{3}}=\sqrt{\frac{5}{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{15}}{3}=\frac{3.88}{3}=1.29\]

**The square root of \[\frac{0.324\times 0.081\times 4.624}{1.5625\times 0.0289\times 72.9\times 64}\]is:**

(a) 24

(b) 2.4

(c) 0.024

(d) 05.69

(e) None of these

** Answer:** (c)

**Explanation**

Simplify the expression by removing the decimals

\[=\frac{324\times 81\times 4624}{15625\times 289\times 729\times 64}=\frac{9}{15625}.\]

\[\therefore \]The square root of \[\frac{9}{15625}=\frac{3}{125}=0.24\]

**Steve gets on the elevator at the eleventh floor of a building and rises up at the rate of 57 floors per minute. At the same time Stephen gets on an elevator at the fifty first floor of the same building and towards down at the rate of 63 floor per minute. If they continue travelling at these rate then at which floor will their elevator cross each other?**

(a) 30

(b) 28

(c) 33

(d) 35

(e) None of these

**Answer:** (a)

**A test tube contains a liquid of red color and another test tube contains an equal amount of water. To make a solution 20 ml of the liquid of red color is poured into the second test tube. Two third of the so formed solution poured from the second into the first. If the liquid in the first test tube is four times that in the second then the quantity of water was taken initially is: **

(a) 30 ml

(b) 50 ml

(c) 30 ml

(d) 60 ml

(e) None of these

**Answer:** (c)

- Fraction is represented as \[\frac{p}{q}\] where p & q are the integers of same sign and \[q\ne 0.\]
- Equivalent fractions are obtained by multiplying its numerator and denominator by same non - zero number.
- A fraction is said to be in lowest form if the common factor of numerator and denominator is not other than 1.
- A fraction is said to be the reciprocal of other if their product is 1.
- For two fractions \[\frac{A}{B},\frac{C}{D}\] if AD > BC then \[\frac{A}{B},>\frac{C}{D}\]. If AD < BC then \[\frac{A}{B}<\frac{C}{D}\]

- \[\frac{1}{1}=1,\,\frac{1}{2}=0.5,\,\,\frac{3}{5}=0.6,\,\frac{3}{25}=0.12\] and \[\frac{1306}{1250}=1.0448.\] Such numbers are the only real numbers which do not have a unique decimal representation, as they can also be written as a representation that has a recurring 9, for instance 1\[=0.99999...,\frac{1}{2}=0.499999...,\]etc.
- The number \[0=\frac{0}{1}\] is special in that it has no representation with recurring 9.

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