Class 9 Number System worksheet with multiple-choice and short-answer questions on rational numbers, reciprocals, and operations with fractions.
A math worksheet for Class 9 Number System from EduGain, featuring questions on rational and irrational numbers, fractions, and decimal conversions.
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Step-by-step solution for: Grade 9 - Number System | Math Practice, Questions, Tests ...
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Show Answer Key & Explanations
Step-by-step solution for: Grade 9 - Number System | Math Practice, Questions, Tests ...
Let’s solve each question one by one, carefully and step by step.
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(1) Find 18 rational numbers between -5/23 and 17/23
Both fractions have the same denominator: 23.
So we can look at numerators: from -5 to 17.
How many integers are between -5 and 17? Let’s count:
From -4 to 16 (inclusive): that’s 16 - (-4) + 1 = 21 numbers.
But we need only 18. So we can pick any 18 of them.
For example:
-4/23, -3/23, -2/23, -1/23, 0/23, 1/23, 2/23, ..., up to 13/23 → that’s 18 numbers.
Wait — let’s check: from -4 to 13 inclusive:
Number of terms = 13 - (-4) + 1 = 18 ✔
So answer:
-4/23, -3/23, -2/23, -1/23, 0, 1/23, 2/23, 3/23, 4/23, 5/23, 6/23, 7/23, 8/23, 9/23, 10/23, 11/23, 12/23, 13/23
*(Note: You can choose any 18 in between — this is just one valid set.)*
---
(2) Find 23 rational numbers between -4/6 and 1/6
First, simplify if needed: -4/6 = -2/3, but since both have denominator 6, keep as is for now.
Numerators: from -4 to 1 → difference is 5 units.
We need 23 numbers between them → so we must use a larger denominator.
Multiply numerator and denominator by 24 (since 23+1=24 intervals needed).
So:
-4/6 = (-4 × 24)/(6 × 24) = -96/144
1/6 = (1 × 24)/(6 × 24) = 24/144
Now find 23 rational numbers between -96/144 and 24/144.
That means numerators from -95 to 23 → total numbers: 23 - (-95) + 1 = 119 → too many.
Actually, we want exactly 23 numbers between them → so we can take numerators from -95 down to... wait, better way:
The gap between -96 and 24 is 120 units. We want 23 numbers → divide into 24 equal parts? Not necessary.
Easier: Just list 23 consecutive fractions with denominator 144 starting from -95/144 up to -73/144? Wait, no — we need to go toward 24/144.
Actually, simplest: Take numerators from -95 to -73? That’s only 23 numbers? Let’s calculate:
From -95 to -73 inclusive: number of terms = (-73) - (-95) + 1 = 23 ✔
But these are all less than -4/6? No — -95/144 ≈ -0.66, while -4/6 ≈ -0.666… so yes, they are greater than -4/6.
Wait — actually, -96/144 = -4/6, so next is -95/144, then -94/144, ..., up to 23/144.
We need 23 numbers → so pick from -95/144 to -73/144? But that’s going away from 1/6.
Better: Pick 23 numbers close to the middle or spread out.
Actually, easiest: Since we have room from -95 to 23 (total 119 numbers), just pick first 23 after -96:
→ -95/144, -94/144, -93/144, ..., down to -73/144? Wait, decreasing numerator makes it more negative.
No: -95 > -96, so -95/144 is greater than -96/144.
To move toward 1/6, we increase numerator.
So start from -95/144 and go up to some point.
We need 23 numbers: so from -95 to -73? Let's compute:
-95, -94, ..., -73 → how many?
Last term = first + (n-1)*step → -73 = -95 + (n-1)(1) → n-1 = 22 → n=23 ✔
But -73/144 is still much less than 24/144 (which is 1/6). So yes, all are between -4/6 and 1/6.
Alternatively, to make it simpler, we could reduce fractions later, but the question doesn’t require lowest terms here.
So answer:
-95/144, -94/144, -93/144, ..., -73/144 (23 numbers)
You can write them as:
-95/144, -47/72, -31/48, ... but better to leave with common denominator unless asked.
Actually, let me double-check: Is -73/144 < 1/6?
1/6 = 24/144 → yes, -73 < 24 → so yes.
And -95/144 > -96/144 → yes.
So correct.
---
(3) Express the following numbers in the form p/q and reduce to lowest terms.
These are repeating decimals.
Recall method: Let x = decimal, multiply by power of 10 to shift repeating part, subtract.
---
A) 0.92675̅ → bar over 75? Or 675? Looking at notation: “0.92675” with bar over last two digits? In text it says “0.92675” — probably bar over 75? But written as “0.92675” — maybe typo? Looking back: “0.92675” — likely bar over 75, meaning 0.926757575...
Assume: 0.92675 with bar over 75 → so non-repeating: 926, repeating: 75? Wait, positions:
Actually, standard notation: if it's written as 0.92675 with bar over 75, it means 0.926757575...
So let x = 0.926757575...
Repeating part has 2 digits, non-repeating after decimal has 3 digits (926).
So multiply by 10^3 = 1000 to get past non-repeating:
1000x = 926.757575...
Multiply by 10^5 = 100000 to shift two full repeats:
100000x = 92675.757575...
Subtract:
100000x - 1000x = 92675.7575... - 926.7575...
→ 99000x = 91749
→ x = 91749 / 99000
Now simplify.
Find GCD of 91749 and 99000.
Use Euclidean algorithm:
GCD(99000, 91749)
99000 ÷ 91749 = 1 rem 7251
91749 ÷ 7251 = 12 rem 4737? Let me compute:
7251 * 12 = 87012
91749 - 87012 = 4737
Then GCD(7251, 4737)
7251 ÷ 4737 = 1 rem 2514
4737 ÷ 2514 = 1 rem 2223
2514 ÷ 2223 = 1 rem 291
2223 ÷ 291 = 7 rem 186? 291*7=2037, 2223-2037=186
291 ÷ 186 = 1 rem 105
186 ÷ 105 = 1 rem 81
105 ÷ 81 = 1 rem 24
81 ÷ 24 = 3 rem 9
24 ÷ 9 = 2 rem 6
9 ÷ 6 = 1 rem 3
6 ÷ 3 = 2 rem 0 → GCD is 3
So divide numerator and denominator by 3:
91749 ÷ 3 = 30583
99000 ÷ 3 = 33000
Check if further divisible? 30583 and 33000.
30583 ÷ 3? 3+0+5+8+3=19 not div by 3.
Ends with 3, not even, not 5. Probably prime? Assume simplified.
So A) 30583/33000
Wait — but let me verify calculation.
Original: x = 0.926757575...
1000x = 926.757575...
100000x = 92675.757575...
Subtract: 99000x = 92675.7575... - 926.7575... = 91749 → yes.
91749 / 99000 ÷3 = 30583/33000
Yes.
---
B) 0.31124̅ — bar over 4? Or 124? Written as “0.31124” — likely bar over last digit 4? So 0.31124444...
Assume: 0.31124 with bar over 4 → so repeating 4.
Let x = 0.31124444...
Non-repeating: 3112 (4 digits), repeating: 4 (1 digit)
Multiply by 10^4 = 10000: 10000x = 3112.4444...
Multiply by 10^5 = 100000: 100000x = 31124.4444...
Subtract: 100000x - 10000x = 31124.444... - 3112.444... = 28012
→ 90000x = 28012
→ x = 28012 / 90000
Simplify.
Divide numerator and denominator by 4:
28012 ÷ 4 = 7003
90000 ÷ 4 = 22500
Check GCD(7003, 22500)
7003 ÷ 22500 <1, so GCD(22500,7003)
22500 ÷ 7003 = 3 rem 1491 (7003*3=21009, 22500-21009=1491)
7003 ÷ 1491 = 4 rem 1039 (1491*4=5964, 7003-5964=1039)
1491 ÷ 1039 = 1 rem 452
1039 ÷ 452 = 2 rem 135
452 ÷ 135 = 3 rem 47
135 ÷ 47 = 2 rem 41
47 ÷ 41 = 1 rem 6
41 ÷ 6 = 6 rem 5
6 ÷ 5 = 1 rem 1
5 ÷ 1 = 5 rem 0 → GCD=1
So B) 7003/22500
---
C) 0.77953̅ — bar over 3? Or 53? Probably bar over 53? Text says “0.77953” — assume bar over 53 → 0.779535353...
Let x = 0.779535353...
Non-repeating: 779 (3 digits), repeating: 53 (2 digits)
Multiply by 10^3 = 1000: 1000x = 779.535353...
Multiply by 10^5 = 100000: 100000x = 77953.535353...
Subtract: 100000x - 1000x = 77953.5353... - 779.5353... = 77174
→ 99000x = 77174
→ x = 77174 / 99000
Simplify.
Divide by 2: 38587 / 49500
Check GCD(38587, 49500)
49500 ÷ 38587 = 1 rem 10913
38587 ÷ 10913 = 3 rem 5848 (10913*3=32739, 38587-32739=5848)
10913 ÷ 5848 = 1 rem 5065
5848 ÷ 5065 = 1 rem 783
5065 ÷ 783 = 6 rem 367 (783*6=4698, 5065-4698=367)
783 ÷ 367 = 2 rem 49
367 ÷ 49 = 7 rem 24
49 ÷ 24 = 2 rem 1
24 ÷ 1 = 24 → GCD=1
So C) 38587/49500
---
D) 0.51867̅ — bar over 7? Or 67? Assume bar over 67 → 0.518676767...
x = 0.518676767...
Non-repeating: 518 (3 digits), repeating: 67 (2 digits)
1000x = 518.676767...
100000x = 51867.676767...
Subtract: 99000x = 51867.6767... - 518.6767... = 51349
→ x = 51349 / 99000
Check simplification.
GCD(51349,99000)
99000 ÷ 51349 = 1 rem 47651
51349 ÷ 47651 = 1 rem 3698
47651 ÷ 3698 = 12 rem 3275? 3698*12=44376, 47651-44376=3275
This is getting messy. Try dividing by small primes.
51349 odd, sum digits 5+1+3+4+9=22 not div by 3. Ends with 9, not 5 or 0. Likely coprime.
Assume 51349/99000
---
E) 0.5803̅ — bar over 3? Or 03? Probably bar over 03 → 0.58030303...
x = 0.58030303...
Non-repeating: 58 (2 digits), repeating: 03 (2 digits)
100x = 58.030303...
10000x = 5803.030303...
Subtract: 9900x = 5803.0303... - 58.0303... = 5745
→ x = 5745 / 9900
Simplify.
Divide by 15: 5745÷15=383, 9900÷15=660? 15*383=5745? 15*380=5700, 15*3=45 → yes. 9900÷15=660.
So 383/660
Check GCD(383,660). 383 is prime? 383÷2,3,5,7,11,13,17,19 — 19*20=380, 383-380=3, not div. So prime. 660 not div by 383. So 383/660
---
F) 0.79430̅ — bar over 0? Or 30? Probably bar over 30 → 0.794303030...
x = 0.794303030...
Non-repeating: 794 (3 digits), repeating: 30 (2 digits)
1000x = 794.303030...
100000x = 79430.303030...
Subtract: 99000x = 79430.3030... - 794.3030... = 78636
→ x = 78636 / 99000
Simplify.
Divide by 12: 78636÷12=6553, 99000÷12=8250? 12*6553=78636? 12*6500=78000, 12*53=636 → yes. 99000÷12=8250.
So 6553/8250
Check GCD(6553,8250)
8250 ÷ 6553 = 1 rem 1697
6553 ÷ 1697 = 3 rem 1462 (1697*3=5091, 6553-5091=1462)
1697 ÷ 1462 = 1 rem 235
1462 ÷ 235 = 6 rem 52 (235*6=1410, 1462-1410=52)
235 ÷ 52 = 4 rem 27
52 ÷ 27 = 1 rem 25
27 ÷ 25 = 1 rem 2
25 ÷ 2 = 12 rem 1
2 ÷ 1 = 2 → GCD=1
So F) 6553/8250
---
(4) Write the rational number that are equal to their reciprocals.
Let the number be x.
Reciprocal is 1/x.
Set x = 1/x → x² = 1 → x = ±1
Both are rational.
So answer: 1 and -1
---
(5) The product of two irrational numbers is _______________
Options:
a. a rational or an irrational number
b. neither rational or irrational number
c. a rational number
d. an irrational number
Counterexamples:
√2 * √2 = 2 → rational
√2 * √3 = 6 → irrational
So sometimes rational, sometimes irrational.
Answer: a. a rational or an irrational number
---
(6) A rational number equivalent to -8/-5 is
-8/-5 = 8/5 (negative divided by negative is positive)
Look at options:
a. -16/10 = -8/5 → not same
b. 16/-10 = -8/5 → not same
c. 16/10 = 8/5 → yes!
d. None of these
So c. 16/10 = 8/5 = -8/-5
Answer: c. 16/10
---
(7) The difference between the greatest and the least numbers of 5/3, 7/2, 4/10, 2/9 is
First, convert to decimals or compare.
5/3 ≈ 1.666...
7/2 = 3.5
4/10 = 0.4
2/9 ≈ 0.222...
Greatest: 7/2 = 3.5
Least: 2/9 ≈ 0.222...
Difference: 7/2 - 2/9
LCM of 2 and 9 is 18.
7/2 = 63/18
2/9 = 4/18
Difference: 63/18 - 4/18 = 59/18
Answer: d. 59/18
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Final Answer:
(1) -4/23, -3/23, -2/23, -1/23, 0, 1/23, 2/23, 3/23, 4/23, 5/23, 6/23, 7/23, 8/23, 9/23, 10/23, 11/23, 12/23, 13/23
(2) -95/144, -94/144, -93/144, ..., -73/144 (23 numbers)
(3) A) 30583/33000, B) 7003/22500, C) 38587/49500, D) 51349/99000, E) 383/660, F) 6553/8250
(4) 1 and -1
(5) a
(6) c
(7) d
---
(1) Find 18 rational numbers between -5/23 and 17/23
Both fractions have the same denominator: 23.
So we can look at numerators: from -5 to 17.
How many integers are between -5 and 17? Let’s count:
From -4 to 16 (inclusive): that’s 16 - (-4) + 1 = 21 numbers.
But we need only 18. So we can pick any 18 of them.
For example:
-4/23, -3/23, -2/23, -1/23, 0/23, 1/23, 2/23, ..., up to 13/23 → that’s 18 numbers.
Wait — let’s check: from -4 to 13 inclusive:
Number of terms = 13 - (-4) + 1 = 18 ✔
So answer:
-4/23, -3/23, -2/23, -1/23, 0, 1/23, 2/23, 3/23, 4/23, 5/23, 6/23, 7/23, 8/23, 9/23, 10/23, 11/23, 12/23, 13/23
*(Note: You can choose any 18 in between — this is just one valid set.)*
---
(2) Find 23 rational numbers between -4/6 and 1/6
First, simplify if needed: -4/6 = -2/3, but since both have denominator 6, keep as is for now.
Numerators: from -4 to 1 → difference is 5 units.
We need 23 numbers between them → so we must use a larger denominator.
Multiply numerator and denominator by 24 (since 23+1=24 intervals needed).
So:
-4/6 = (-4 × 24)/(6 × 24) = -96/144
1/6 = (1 × 24)/(6 × 24) = 24/144
Now find 23 rational numbers between -96/144 and 24/144.
That means numerators from -95 to 23 → total numbers: 23 - (-95) + 1 = 119 → too many.
Actually, we want exactly 23 numbers between them → so we can take numerators from -95 down to... wait, better way:
The gap between -96 and 24 is 120 units. We want 23 numbers → divide into 24 equal parts? Not necessary.
Easier: Just list 23 consecutive fractions with denominator 144 starting from -95/144 up to -73/144? Wait, no — we need to go toward 24/144.
Actually, simplest: Take numerators from -95 to -73? That’s only 23 numbers? Let’s calculate:
From -95 to -73 inclusive: number of terms = (-73) - (-95) + 1 = 23 ✔
But these are all less than -4/6? No — -95/144 ≈ -0.66, while -4/6 ≈ -0.666… so yes, they are greater than -4/6.
Wait — actually, -96/144 = -4/6, so next is -95/144, then -94/144, ..., up to 23/144.
We need 23 numbers → so pick from -95/144 to -73/144? But that’s going away from 1/6.
Better: Pick 23 numbers close to the middle or spread out.
Actually, easiest: Since we have room from -95 to 23 (total 119 numbers), just pick first 23 after -96:
→ -95/144, -94/144, -93/144, ..., down to -73/144? Wait, decreasing numerator makes it more negative.
No: -95 > -96, so -95/144 is greater than -96/144.
To move toward 1/6, we increase numerator.
So start from -95/144 and go up to some point.
We need 23 numbers: so from -95 to -73? Let's compute:
-95, -94, ..., -73 → how many?
Last term = first + (n-1)*step → -73 = -95 + (n-1)(1) → n-1 = 22 → n=23 ✔
But -73/144 is still much less than 24/144 (which is 1/6). So yes, all are between -4/6 and 1/6.
Alternatively, to make it simpler, we could reduce fractions later, but the question doesn’t require lowest terms here.
So answer:
-95/144, -94/144, -93/144, ..., -73/144 (23 numbers)
You can write them as:
-95/144, -47/72, -31/48, ... but better to leave with common denominator unless asked.
Actually, let me double-check: Is -73/144 < 1/6?
1/6 = 24/144 → yes, -73 < 24 → so yes.
And -95/144 > -96/144 → yes.
So correct.
---
(3) Express the following numbers in the form p/q and reduce to lowest terms.
These are repeating decimals.
Recall method: Let x = decimal, multiply by power of 10 to shift repeating part, subtract.
---
A) 0.92675̅ → bar over 75? Or 675? Looking at notation: “0.92675” with bar over last two digits? In text it says “0.92675” — probably bar over 75? But written as “0.92675” — maybe typo? Looking back: “0.92675” — likely bar over 75, meaning 0.926757575...
Assume: 0.92675 with bar over 75 → so non-repeating: 926, repeating: 75? Wait, positions:
Actually, standard notation: if it's written as 0.92675 with bar over 75, it means 0.926757575...
So let x = 0.926757575...
Repeating part has 2 digits, non-repeating after decimal has 3 digits (926).
So multiply by 10^3 = 1000 to get past non-repeating:
1000x = 926.757575...
Multiply by 10^5 = 100000 to shift two full repeats:
100000x = 92675.757575...
Subtract:
100000x - 1000x = 92675.7575... - 926.7575...
→ 99000x = 91749
→ x = 91749 / 99000
Now simplify.
Find GCD of 91749 and 99000.
Use Euclidean algorithm:
GCD(99000, 91749)
99000 ÷ 91749 = 1 rem 7251
91749 ÷ 7251 = 12 rem 4737? Let me compute:
7251 * 12 = 87012
91749 - 87012 = 4737
Then GCD(7251, 4737)
7251 ÷ 4737 = 1 rem 2514
4737 ÷ 2514 = 1 rem 2223
2514 ÷ 2223 = 1 rem 291
2223 ÷ 291 = 7 rem 186? 291*7=2037, 2223-2037=186
291 ÷ 186 = 1 rem 105
186 ÷ 105 = 1 rem 81
105 ÷ 81 = 1 rem 24
81 ÷ 24 = 3 rem 9
24 ÷ 9 = 2 rem 6
9 ÷ 6 = 1 rem 3
6 ÷ 3 = 2 rem 0 → GCD is 3
So divide numerator and denominator by 3:
91749 ÷ 3 = 30583
99000 ÷ 3 = 33000
Check if further divisible? 30583 and 33000.
30583 ÷ 3? 3+0+5+8+3=19 not div by 3.
Ends with 3, not even, not 5. Probably prime? Assume simplified.
So A) 30583/33000
Wait — but let me verify calculation.
Original: x = 0.926757575...
1000x = 926.757575...
100000x = 92675.757575...
Subtract: 99000x = 92675.7575... - 926.7575... = 91749 → yes.
91749 / 99000 ÷3 = 30583/33000
Yes.
---
B) 0.31124̅ — bar over 4? Or 124? Written as “0.31124” — likely bar over last digit 4? So 0.31124444...
Assume: 0.31124 with bar over 4 → so repeating 4.
Let x = 0.31124444...
Non-repeating: 3112 (4 digits), repeating: 4 (1 digit)
Multiply by 10^4 = 10000: 10000x = 3112.4444...
Multiply by 10^5 = 100000: 100000x = 31124.4444...
Subtract: 100000x - 10000x = 31124.444... - 3112.444... = 28012
→ 90000x = 28012
→ x = 28012 / 90000
Simplify.
Divide numerator and denominator by 4:
28012 ÷ 4 = 7003
90000 ÷ 4 = 22500
Check GCD(7003, 22500)
7003 ÷ 22500 <1, so GCD(22500,7003)
22500 ÷ 7003 = 3 rem 1491 (7003*3=21009, 22500-21009=1491)
7003 ÷ 1491 = 4 rem 1039 (1491*4=5964, 7003-5964=1039)
1491 ÷ 1039 = 1 rem 452
1039 ÷ 452 = 2 rem 135
452 ÷ 135 = 3 rem 47
135 ÷ 47 = 2 rem 41
47 ÷ 41 = 1 rem 6
41 ÷ 6 = 6 rem 5
6 ÷ 5 = 1 rem 1
5 ÷ 1 = 5 rem 0 → GCD=1
So B) 7003/22500
---
C) 0.77953̅ — bar over 3? Or 53? Probably bar over 53? Text says “0.77953” — assume bar over 53 → 0.779535353...
Let x = 0.779535353...
Non-repeating: 779 (3 digits), repeating: 53 (2 digits)
Multiply by 10^3 = 1000: 1000x = 779.535353...
Multiply by 10^5 = 100000: 100000x = 77953.535353...
Subtract: 100000x - 1000x = 77953.5353... - 779.5353... = 77174
→ 99000x = 77174
→ x = 77174 / 99000
Simplify.
Divide by 2: 38587 / 49500
Check GCD(38587, 49500)
49500 ÷ 38587 = 1 rem 10913
38587 ÷ 10913 = 3 rem 5848 (10913*3=32739, 38587-32739=5848)
10913 ÷ 5848 = 1 rem 5065
5848 ÷ 5065 = 1 rem 783
5065 ÷ 783 = 6 rem 367 (783*6=4698, 5065-4698=367)
783 ÷ 367 = 2 rem 49
367 ÷ 49 = 7 rem 24
49 ÷ 24 = 2 rem 1
24 ÷ 1 = 24 → GCD=1
So C) 38587/49500
---
D) 0.51867̅ — bar over 7? Or 67? Assume bar over 67 → 0.518676767...
x = 0.518676767...
Non-repeating: 518 (3 digits), repeating: 67 (2 digits)
1000x = 518.676767...
100000x = 51867.676767...
Subtract: 99000x = 51867.6767... - 518.6767... = 51349
→ x = 51349 / 99000
Check simplification.
GCD(51349,99000)
99000 ÷ 51349 = 1 rem 47651
51349 ÷ 47651 = 1 rem 3698
47651 ÷ 3698 = 12 rem 3275? 3698*12=44376, 47651-44376=3275
This is getting messy. Try dividing by small primes.
51349 odd, sum digits 5+1+3+4+9=22 not div by 3. Ends with 9, not 5 or 0. Likely coprime.
Assume 51349/99000
---
E) 0.5803̅ — bar over 3? Or 03? Probably bar over 03 → 0.58030303...
x = 0.58030303...
Non-repeating: 58 (2 digits), repeating: 03 (2 digits)
100x = 58.030303...
10000x = 5803.030303...
Subtract: 9900x = 5803.0303... - 58.0303... = 5745
→ x = 5745 / 9900
Simplify.
Divide by 15: 5745÷15=383, 9900÷15=660? 15*383=5745? 15*380=5700, 15*3=45 → yes. 9900÷15=660.
So 383/660
Check GCD(383,660). 383 is prime? 383÷2,3,5,7,11,13,17,19 — 19*20=380, 383-380=3, not div. So prime. 660 not div by 383. So 383/660
---
F) 0.79430̅ — bar over 0? Or 30? Probably bar over 30 → 0.794303030...
x = 0.794303030...
Non-repeating: 794 (3 digits), repeating: 30 (2 digits)
1000x = 794.303030...
100000x = 79430.303030...
Subtract: 99000x = 79430.3030... - 794.3030... = 78636
→ x = 78636 / 99000
Simplify.
Divide by 12: 78636÷12=6553, 99000÷12=8250? 12*6553=78636? 12*6500=78000, 12*53=636 → yes. 99000÷12=8250.
So 6553/8250
Check GCD(6553,8250)
8250 ÷ 6553 = 1 rem 1697
6553 ÷ 1697 = 3 rem 1462 (1697*3=5091, 6553-5091=1462)
1697 ÷ 1462 = 1 rem 235
1462 ÷ 235 = 6 rem 52 (235*6=1410, 1462-1410=52)
235 ÷ 52 = 4 rem 27
52 ÷ 27 = 1 rem 25
27 ÷ 25 = 1 rem 2
25 ÷ 2 = 12 rem 1
2 ÷ 1 = 2 → GCD=1
So F) 6553/8250
---
(4) Write the rational number that are equal to their reciprocals.
Let the number be x.
Reciprocal is 1/x.
Set x = 1/x → x² = 1 → x = ±1
Both are rational.
So answer: 1 and -1
---
(5) The product of two irrational numbers is _______________
Options:
a. a rational or an irrational number
b. neither rational or irrational number
c. a rational number
d. an irrational number
Counterexamples:
√2 * √2 = 2 → rational
√2 * √3 = 6 → irrational
So sometimes rational, sometimes irrational.
Answer: a. a rational or an irrational number
---
(6) A rational number equivalent to -8/-5 is
-8/-5 = 8/5 (negative divided by negative is positive)
Look at options:
a. -16/10 = -8/5 → not same
b. 16/-10 = -8/5 → not same
c. 16/10 = 8/5 → yes!
d. None of these
So c. 16/10 = 8/5 = -8/-5
Answer: c. 16/10
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(7) The difference between the greatest and the least numbers of 5/3, 7/2, 4/10, 2/9 is
First, convert to decimals or compare.
5/3 ≈ 1.666...
7/2 = 3.5
4/10 = 0.4
2/9 ≈ 0.222...
Greatest: 7/2 = 3.5
Least: 2/9 ≈ 0.222...
Difference: 7/2 - 2/9
LCM of 2 and 9 is 18.
7/2 = 63/18
2/9 = 4/18
Difference: 63/18 - 4/18 = 59/18
Answer: d. 59/18
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Final Answer:
(1) -4/23, -3/23, -2/23, -1/23, 0, 1/23, 2/23, 3/23, 4/23, 5/23, 6/23, 7/23, 8/23, 9/23, 10/23, 11/23, 12/23, 13/23
(2) -95/144, -94/144, -93/144, ..., -73/144 (23 numbers)
(3) A) 30583/33000, B) 7003/22500, C) 38587/49500, D) 51349/99000, E) 383/660, F) 6553/8250
(4) 1 and -1
(5) a
(6) c
(7) d
Parent Tip: Review the logic above to help your child master the concept of number system worksheet.
