10th Maths 1.4
NCERT Class 10th solution of Exercise 1.1
NCERT Class 10th solution of Exercise 1.2
NCERT Class 10th solution of Exercise 1.3
Exercise 1.4
Q1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion :
i) `13/3125`
Sol. :
`13/3125=13/5^5`
Since the denominator is in the form `2^n.5^m,` where `n` and `m` are non-negative integers.
Answer :
Thus, the decimal expansion is terminating.
ii) `17/8`
Sol. :
`17/8=17/2^3`
Since the denominator is in the form `2^n.5^m,` where `n` and `m` are non-negative integers.
Answer :
Thus, the decimal expansion is terminating.
iii) `64/455`
Sol. :
`64/455=64/(5^1times7^1times13^1)`
Since the denominator has `7` and `13` also other than `2` and `5` as a prime factor.
Answer :
Thus, the decimal expansion is non-terminating recurring.
iv) `15/1600`
Sol. :
`15/6000=(5times3)/(2^6times5^2)=3/(2^6times5)`
Since the denominator is in the form `2^n.5^m,` where `n` and `m` are non-negative integers.
Answer :
Thus, the decimal expansion is terminating.
v) `29/343`
Sol. :
`29/343=29/7^3`
Since the denominator has the power of `7` other than `2` and `5` as a prime fator.
Answer :
Thus, the decimal expansion is non-terminating recurring.
vi) `23/(2³5²)`
Sol. :
`23/(2³5²)=23/(2³5²)`
Since the denominator is in the form `2^n.5^m,` where `n` and `m` are non-negative integers.
Answer :
Thus, the decimal expansion is terminating.
vii) `129/(2²5⁷7⁵)`
Sol. :
`129/(2²5⁷7⁵)`
Since the denominator has the power of `7` other than `2^n`and `5^m`.
Answer :
Thus, the decimal expansion is non-terminating recurring.
viii) `6/15`
Sol. :
`6/15=2/5`
Since the denominator is in the form `2^n.5^m,` where `n` and `m` are non-negative integers.
Answer :
Thus, the decimal expansion is terminating.
ix) `35/50`
Sol.
`35/50=7/10=7/(2times5)`
Since the denominator is in the form `2^n.5^m,` where `n` and `m` are non-negative integers.
Answer :
Thus, the decimal expansion is terminating.
x) `77/210`
Sol. :
`77/210=11/30=11/(2times3times5)`
Since the denominator has `3` also other than `2^n.5^m`.
Answer :
Thus, the decimal expansion is non-terminating recurring.
Q2. Write down the decimal expansions of those rational numbers in Question `1` above which have terminating decimal expansions.
Sol.
We solve those rational numbers in Question `1` above which have terminating decimal expansions
i)
`13/3125`
`=(13times2^5)/(5^5times2^5)`
`=416/10^5`
`=0.00416`
Answer :
`=0.00416`
ii)
`17/8`
`=17/2^3`
`=(17times5^3)/(2^3times5^3)`
`=2125/10^3`
`=2.125`
Answer :
`=2.125`
iii)
`15/1600`
`=(3times5)/(2^6times5^2)`
`=(3times5times5^4)/(2^6times5^2times5^4)`
`=9375/10^6`
`=0.009375`
Answer :
`=0.009375`
iv)
`23/(2^3times5^2)`
`=(23times5)/(2^3times5^2times5)`
`=115/10^3`
`=0.115`
Answer :
`=0.115`
v)
`6/15`
`=(2times3)/(5times3)`
`=(2times2)/(5times2)`
`=4/10`
`=0.4`
Answer :
`=0.4`
vi)
`35/50`
`=(7times5)/(2times5^2)`
`=7/(2times5)`
`=7/10`
`=0.7`
Answer :
`=0.7`
Q3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form `p/q`, what can you say about the prime factors of `q`.
i) `43.123456789` ii) `0.120120012000120000`.. iii) `43.overline123456789`
Sol. :
`43.123456789`
`=43123456789/10^9`
`=p/q`
Since the decimal expansion is terminating.
Answer :
Thus, the given number is a rational number that can be expressed in the form `p/q` and the prime factors of `q` are `2^n.5^m`.
ii)
`0.120120012000120000`..
Since the given decimal expansion is non-terminating non-recurring.
Answer :
Thus, the number is an irrational number.
iii)
`43.overline123456789`
Since the given decimal expansion is non-terminating recurring.
Answer :
Thus, the number is rational and is in the form `p/q` in which q has prime factors other than `2^n.5^m` also.
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