Table of Contents
Exercise 1.3 Solutions – Class X Mathematics
These solutions are based on the Telangana State Class X Mathematics textbook, focusing on decimal expansions and rational numbers. Mathematical expressions are rendered using MathJax.
1. Write the following rational numbers in their decimal form and also state which are terminating and which are non-terminating, repeating decimal.
(i) \( \frac{3}{8} \)
\( \frac{3}{8} = 3 \div 8 = 0.375 \) (long division)
Denominator \( 8 = 2^3 \), only 2 as factor, so terminating.
Decimal = 0.375, Terminating
(ii) \( \frac{229}{400} \)
\( \frac{229}{400} = 229 \div 400 = 0.5725 \) (long division)
Denominator \( 400 = 2^4 \times 5^2 \), only 2 and 5, so terminating.
Decimal = 0.5725, Terminating
(iii) \( \frac{4}{5} \)
\( \frac{4}{5} = 4 \div 5 = 0.8 \) (long division)
Denominator \( 5 = 5^1 \), only 5, so terminating.
Decimal = 0.8, Terminating
(iv) \( \frac{2}{11} \)
\( \frac{2}{11} = 2 \div 11 = 0.181818\ldots \) (long division)
Repeating digit: 18
Denominator \( 11 \) has factor other than 2 or 5, so non-terminating repeating.
Decimal = 0.\overline{18}, Non-terminating repeating
(v) \( \frac{8}{125} \)
\( \frac{8}{125} = 8 \div 125 = 0.064 \) (long division)
Denominator \( 125 = 5^3 \), only 5, so terminating.
Decimal = 0.064, Terminating
2. Without performing division, state whether the following rational numbers will have a terminating decimal form or a non-terminating, repeating decimal form.
A rational number \( \frac{p}{q} \) (in lowest form) has a terminating decimal if \( q = 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers.
(i) \( \frac{13}{3125} \)
Denominator \( 3125 = 5^5 \)
Only 5 as factor, so terminating.
Terminating
(ii) \( \frac{15}{16} \)
Denominator \( 16 = 2^4 \)
Only 2 as factor, so terminating.
Terminating
(iii) \( \frac{23}{2^3 \cdot 5^2} \)
Denominator \( 2^3 \times 5^2 = 8 \times 25 = 200 \)
Only 2 and 5 as factors, so terminating.
Terminating
(iv) \( \frac{7218}{3^2 \cdot 5^2} \)
Denominator \( 3^2 \times 5^2 = 9 \times 25 = 225 \)
Has 3 as factor (not just 2 or 5), so non-terminating repeating.
Non-terminating repeating
(v) \( \frac{143}{110} \)
Denominator \( 110 = 2 \times 5 \times 11 \)
Has 11 as factor (not just 2 or 5), so non-terminating repeating.
Non-terminating repeating
(vi) \( \frac{23}{2^3 \cdot 5^2} \)
Denominator \( 2^3 \times 5^2 = 8 \times 25 = 200 \)
Only 2 and 5 as factors, so terminating.
Terminating
(vii) \( \frac{129}{2^2 \cdot 5^2 \cdot 7^2} \)
Denominator \( 2^2 \times 5^2 \times 7^2 = 4 \times 25 \times 49 = 4900 \)
Has 7 as factor (not just 2 or 5), so non-terminating repeating.
Non-terminating repeating
(viii) \( \frac{9}{15} \)
Denominator \( 15 = 3 \times 5 \)
Has 3 as factor (not just 2 or 5), so non-terminating repeating.
Non-terminating repeating
(ix) \( \frac{36}{100} \)
Denominator \( 100 = 2^2 \times 5^2 \)
Only 2 and 5 as factors, so terminating.
Terminating
(x) \( \frac{77}{210} \)
Denominator \( 210 = 2 \times 3 \times 5 \times 7 \)
Has 3 and 7 as factors (not just 2 or 5), so non-terminating repeating.
Non-terminating repeating
3. Write the following rationals in decimal form using Theorem 1.4.
(i) \( \frac{13}{25} \)
Denominator \( 25 = 5^2 \)
Express with power of 10: \( \frac{13}{25} = \frac{13 \times 4}{25 \times 4} = \frac{52}{100} = 0.52 \)
Terminating decimal.
Decimal = 0.52
(ii) \( \frac{15}{16} \)
Denominator \( 16 = 2^4 \)
Express with power of 10: \( \frac{15}{16} = \frac{15 \times 625}{16 \times 625} = \frac{9375}{10000} = 0.9375 \)
Terminating decimal.
Decimal = 0.9375
(iii) \( \frac{23}{2^3 \cdot 5^2} \)
Denominator \( 2^3 \times 5^2 = 8 \times 25 = 200 \)
Express with power of 10: \( \frac{23}{200} = \frac{23 \times 5}{200 \times 5} = \frac{115}{1000} = 0.115 \)
Terminating decimal.
Decimal = 0.115
(iv) \( \frac{7218}{3^2 \cdot 5^2} \)
Denominator \( 3^2 \times 5^2 = 9 \times 25 = 225 \)
Has 3, cannot be expressed as \( 2^m \times 5^n \) alone, so non-terminating repeating.
Approximate: \( 7218 \div 225 \approx 32.08 \) (repeating).
Decimal = 32.08\overline{…}, Non-terminating repeating
(v) \( \frac{143}{110} \)
Denominator \( 110 = 2 \times 5 \times 11 \)
Has 11, cannot be expressed as \( 2^m \times 5^n \) alone, so non-terminating repeating.
Approximate: \( 143 \div 110 \approx 1.3 \) (repeating).
Decimal = 1.3\overline{…}, Non-terminating repeating
4. Express the following decimals in the form of \( \frac{p}{q} \), and write the prime factors of \( q \). What do you observe?
(i) 43.123
Let \( x = 43.123 \)
\( 1000x = 43123.123 \)
\( 1000x – x = 43123.123 – 43.123 \)
\( 999x = 43080 \)
\( x = \frac{43080}{999} \)
Prime factors of \( 999 = 3^3 \times 37 \)
Observation: Denominator has factors other than 2 and 5, indicating non-terminating repeating.
Form = \(\frac{43080}{999}\), Prime factors of \( q = 3^3 \times 37\)
(ii) 0.1201201
Let \( x = 0.1201201 \)
\( 1000000x = 120120.1 \)
\( 1000x = 120.1201 \)
\( 1000000x – 1000x = 120120.1 – 120.1201 \)
\( 999000x = 120000 \)
\( x = \frac{120000}{999000} = \frac{4}{33} \) (simplify)
Prime factors of \( 33 = 3 \times 11 \)
Observation: Denominator has factors other than 2 and 5, indicating non-terminating repeating.
Form = \(\frac{4}{33}\), Prime factors of \( q = 3 \times 11\)
(iii) 43.12
Let \( x = 43.12 \)
\( 100x = 4312.12 \)
\( 100x – x = 4312.12 – 43.12 \)
\( 99x = 4269 \)
\( x = \frac{4269}{99} = \frac{1423}{33} \) (simplify)
Prime factors of \( 33 = 3 \times 11 \)
Observation: Denominator has factors other than 2 and 5, indicating non-terminating repeating.
Form = \(\frac{1423}{33}\), Prime factors of \( q = 3 \times 11\)
(iv) 0.63
Let \( x = 0.63 \)
\( 100x = 63.63 \)
\( 100x – x = 63.63 – 0.63 \)
\( 99x = 63 \)
\( x = \frac{63}{99} = \frac{7}{11} \) (simplify)
Prime factors of \( 11 = 11 \)
Observation: Denominator has factors other than 2 and 5, indicating non-terminating repeating.
Form = \(\frac{7}{11}\), Prime factors of \( q = 11\)
