Class 9 Number System worksheet with questions on rational and irrational numbers.

Here are the step-by-step solutions for each question on the worksheet.

Answer the questions

(1) Find 18 rational numbers between $\frac{-5}{23}$ and $\frac{17}{23}$.

Since the denominators are already the same (23), we just need to look at the numerators. We need to find integers between -5 and 17.
The integers between -5 and 17 are:
-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16.

There are 21 numbers here. We only need 18. We can pick any 18 from this list. Let's pick the first 18 starting from -4.

Answer:
$\frac{-4}{23}, \frac{-3}{23}, \frac{-2}{23}, \frac{-1}{23}, \frac{0}{23}, \frac{1}{23}, \frac{2}{23}, \frac{3}{23}, \frac{4}{23}, \frac{5}{23}, \frac{6}{23}, \frac{7}{23}, \frac{8}{23}, \frac{9}{23}, \frac{10}{23}, \frac{11}{23}, \frac{12}{23}, \frac{13}{23}$

*(Note: Any 18 distinct fractions with denominator 23 and numerators between -5 and 17 are correct.)*

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(2) Find 23 rational numbers between $\frac{-4}{6}$ and $\frac{1}{6}$.

The denominators are the same (6). The numerators are -4 and 1.
The integers between -4 and 1 are: -3, -2, -1, 0.
There are only 4 numbers here, but we need 23. To get more numbers, we must increase the denominator by multiplying both fractions by a larger number.

Let's multiply the numerator and denominator by 10 to make the gap bigger.
$\frac{-4 \times 10}{6 \times 10} = \frac{-40}{60}$
$\frac{1 \times 10}{6 \times 10} = \frac{10}{60}$

Now we need 23 numbers between $\frac{-40}{60}$ and $\frac{10}{60}$.
The total count of integers between -40 and 10 is $10 - (-40) - 1 = 49$ numbers. This is plenty. We just need to list 23 of them. Let's start from -39 and go up.

Answer:
$\frac{-39}{60}, \frac{-38}{60}, \frac{-37}{60}, \dots, \frac{-17}{60}$
*(You can list any 23 fractions with denominator 60 where the numerator is between -40 and 10. For example: $\frac{-39}{60}, \frac{-38}{60}, \frac{-37}{60}, \frac{-36}{60}, \frac{-35}{60}, \frac{-34}{60}, \frac{-33}{60}, \frac{-32}{60}, \frac{-31}{60}, \frac{-30}{60}, \frac{-29}{60}, \frac{-28}{60}, \frac{-27}{60}, \frac{-26}{60}, \frac{-25}{60}, \frac{-24}{60}, \frac{-23}{60}, \frac{-22}{60}, \frac{-21}{60}, \frac{-20}{60}, \frac{-19}{60}, \frac{-18}{60}, \frac{-17}{60}$)*

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(3) Express the following numbers in the form of $\frac{p}{q}$ and reduce it to the lowest terms.

To convert repeating decimals to fractions, we use algebra. Let $x$ be the number.

A) $0.92\overline{675}$
This has non-repeating part "92" and repeating part "675".
$x = 0.92675675\dots$
Multiply by $10^5$ (5 decimal places total): $100000x = 92675.675675\dots$
Multiply by $10^2$ (2 non-repeating places): $100x = 92.675675\dots$
Subtract the second from the first:
$99900x = 92675 - 92 = 92583$
$x = \frac{92583}{99900}$
Both are divisible by 9 (sum of digits for top: $9+2+5+8+3=27$, bottom: $9+9+9=27$).
$92583 \div 9 = 10287$
$99900 \div 9 = 11100$
Check divisibility by 3 again ($1+0+2+8+7=18$, $1+1+1=3$).
$10287 \div 3 = 3429$
$11100 \div 3 = 3700$
3429 is not divisible by 2 or 5. Sum of digits $3+4+2+9=18$ (div by 9), but 3700 is not div by 3.
So, simplest form is $\frac{3429}{3700}$.

B) $0.31\overline{124}$
Non-repeating: "31", Repeating: "124". Total 5 decimal places.
$x = 0.31124124\dots$
$100000x = 31124.124124\dots$
$100x = 31.124124\dots$
$99900x = 31124 - 31 = 31093$
$x = \frac{31093}{99900}$
31093 is not divisible by 2, 5, or 3 (sum=16). 99900 is divisible by many things, but since the numerator is prime to small factors, let's check if they share any large factors. It turns out 31093 is likely prime or shares no common factors with 99900.
Answer: $\frac{31093}{99900}$

C) $0.77\overline{953}$
Non-repeating: "77", Repeating: "953".
$100000x = 77953.953953\dots$
$100x = 77.953953\dots$
$99900x = 77953 - 77 = 77876$
$x = \frac{77876}{99900}$
Divide by 4:
$77876 \div 4 = 19469$
$99900 \div 4 = 24975$
19469 is odd. Sum digits $1+9+4+6+9=29$ (not div by 3). Ends in 9 (not 5).
Answer: $\frac{19469}{24975}$

D) $0.51\overline{867}$
Non-repeating: "51", Repeating: "867".
$100000x = 51867.867867\dots$
$100x = 51.867867\dots$
$99900x = 51867 - 51 = 51816$
$x = \frac{51816}{99900}$
Divide by 12 (both div by 4 and 3):
$51816 \div 12 = 4318$
$99900 \div 12 = 8325$
4318 is even, 8325 is odd. No more 2s.
Sum 4318: $4+3+1+8=16$ (no 3).
Ends in 8/5 (no 5).
Answer: $\frac{4318}{8325}$

E) $0.5\overline{803}$
Non-repeating: "5", Repeating: "803".
$10000x = 5803.803803\dots$
$10x = 5.803803\dots$
$9990x = 5803 - 5 = 5798$
$x = \frac{5798}{9990}$
Divide by 2:
$\frac{2899}{4995}$
2899 sum=28 (no 3). Ends in 9.
4995 ends in 5.
No obvious common factors.
Answer: $\frac{2899}{4995}$

F) $0.79\overline{430}$
Non-repeating: "79", Repeating: "430".
$100000x = 79430.430430\dots$
$100x = 79.430430\dots$
$99900x = 79430 - 79 = 79351$
$x = \frac{79351}{99900}$
79351 is not div by 2, 5, 3 (sum=25).
Answer: $\frac{79351}{99900}$

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(4) Write the rational number that are equal to their reciprocals.

Let the number be $x$. Its reciprocal is $\frac{1}{x}$.
We want $x = \frac{1}{x}$.
Multiply by $x$: $x^2 = 1$.
$x = 1$ or $x = -1$.

Answer:
$1$ and $-1$

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Choose correct answer(s) from given choice

(5) The product of two irrational numbers is \_\_\_\_\_\_\_\_\_\_.

* Example 1: $\sqrt{2} \times \sqrt{2} = 2$ (Rational)
* Example 2: $\sqrt{2} \times \sqrt{3} = \sqrt{6}$ (Irrational)

Since the result can be either rational or irrational depending on the numbers chosen, the correct description covers both possibilities.

Correct Choice: a. a rational or an irrational number

(6) A rational number equivalent to $\frac{-8}{-5}$ is

First, simplify the given fraction. A negative divided by a negative is positive.
$\frac{-8}{-5} = \frac{8}{5}$

Now check the options:
a. $\frac{-16}{10} = -1.6$ (Negative)
b. $\frac{16}{-10} = -1.6$ (Negative)
c. $\frac{16}{10}$. Divide top and bottom by 2: $\frac{8}{5} = 1.6$ (Positive and matches)

Correct Choice: c. $\frac{16}{10}$

(7) The difference between the greatest and the least numbers of $\frac{5}{3}, \frac{7}{2}, \frac{4}{10}, \frac{2}{9}$ is

First, find the decimal value or common denominator to compare them.
* $\frac{5}{3} \approx 1.66$
* $\frac{7}{2} = 3.5$
* $\frac{4}{10} = 0.4$
* $\frac{2}{9} \approx 0.22$

Greatest number: $\frac{7}{2}$
Least number: $\frac{2}{9}$

Difference = Greatest - Least
$= \frac{7}{2} - \frac{2}{9}$

Find common denominator for 2 and 9, which is 18.
$\frac{7 \times 9}{18} - \frac{2 \times 2}{18}$
$= \frac{63}{18} - \frac{4}{18}$
$= \frac{59}{18}$

Correct Choice: d. $\frac{59}{18}$

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Final Answer:
(1) $\frac{-4}{23}, \frac{-3}{23}, \frac{-2}{23}, \frac{-1}{23}, \frac{0}{23}, \frac{1}{23}, \frac{2}{23}, \frac{3}{23}, \frac{4}{23}, \frac{5}{23}, \frac{6}{23}, \frac{7}{23}, \frac{8}{23}, \frac{9}{23}, \frac{10}{23}, \frac{11}{23}, \frac{12}{23}, \frac{13}{23}$
(2) $\frac{-39}{60}, \frac{-38}{60}, \frac{-37}{60}, \frac{-36}{60}, \frac{-35}{60}, \frac{-34}{60}, \frac{-33}{60}, \frac{-32}{60}, \frac{-31}{60}, \frac{-30}{60}, \frac{-29}{60}, \frac{-28}{60}, \frac{-27}{60}, \frac{-26}{60}, \frac{-25}{60}, \frac{-24}{60}, \frac{-23}{60}, \frac{-22}{60}, \frac{-21}{60}, \frac{-20}{60}, \frac{-19}{60}, \frac{-18}{60}, \frac{-17}{60}$
(3) A) $\frac{3429}{3700}$, B) $\frac{31093}{99900}$, C) $\frac{19469}{24975}$, D) $\frac{4318}{8325}$, E) $\frac{2899}{4995}$, F) $\frac{79351}{99900}$
(4) $1, -1$
(5) a
(6) c
(7) d