Class 7 Maths - Rational Numbers NCERT Solutions

Chapter 9: Rational Numbers (NCERT Solutions)

Exercise 9.1

Q1. List five rational numbers between:
(i) -1 and 0 (ii) -2 and -1 (iii) -4/5 and -2/3 (iv) -1/2 and 2/3

(i) -1 and 0
Convert to fractions with denominator 6: -1 = -6/6 and 0 = 0/6
Five rational numbers: -5/6, -4/6, -3/6, -2/6, -1/6

(ii) -2 and -1
Convert to fractions with denominator 6: -2 = -12/6 and -1 = -6/6
Five rational numbers: -11/6, -10/6, -9/6, -8/6, -7/6

(iii) -4/5 and -2/3
LCM of 5 and 3 is 15. Make denominators equal: -12/15 and -10/15.
Multiply both by 3/3 to widen the gap: -36/45 and -30/45.
Five rational numbers: -35/45, -34/45, -33/45, -32/45, -31/45

(iv) -1/2 and 2/3
LCM of 2 and 3 is 6. Make denominators equal: -3/6 and 4/6.
Five rational numbers: -2/6, -1/6, 0/6, 1/6, 2/6

Q2. Write four more rational numbers in each of the following patterns:
(i) -3/5, -6/10, -9/15, -12/20, ...
(ii) -1/4, -2/8, -3/12, ...
(iii) -1/6, 2/-12, 3/-18, 4/-24, ...
(iv) -2/3, 2/-3, 4/-6, 6/-9, ...

(i) -3/5, -6/10, -9/15, -12/20, ...
Pattern: Multiplied by 1, 2, 3, 4...
Next four: -3×5 / 5×5, -3×6 / 5×6, -3×7 / 5×7, -3×8 / 5×8
= -15/25, -18/30, -21/35, -24/40

(ii) -1/4, -2/8, -3/12, ...
Pattern: Multiplied by 1, 2, 3...
Next four: -1×4 / 4×4, -1×5 / 4×5, -1×6 / 4×6, -1×7 / 4×7
= -4/16, -5/20, -6/24, -7/28

(iii) -1/6, 2/-12, 3/-18, 4/-24, ...
Pattern is equivalent to multiplying 1/-6 by 1, 2, 3, 4...
Next four: 5/-30, 6/-36, 7/-42, 8/-48
= 5/-30, 6/-36, 7/-42, 8/-48

(iv) -2/3, 2/-3, 4/-6, 6/-9, ...
Note: 2/-3 is same as -2/3. Pattern: 2/-3 × 1, 2, 3...
Next four: 2×4 / -3×4, 2×5 / -3×5, 2×6 / -3×6, 2×7 / -3×7
= 8/-12, 10/-15, 12/-18, 14/-21

Q3. Give four rational numbers equivalent to:
(i) -2/7 (ii) 5/-3 (iii) 4/9

Multiply numerator and denominator by 2, 3, 4, 5:

(i) -2/7 = -4/14, -6/21, -8/28, -10/35

(ii) 5/-3 = 10/-6, 15/-9, 20/-12, 25/-15

(iii) 4/9 = 8/18, 12/27, 16/36, 20/45

Q4. Draw the number line and represent the following rational numbers on it:
(i) 3/4 (ii) -5/8 (iii) -7/4 (iv) 7/8

(Visual representation via text logic)

(i) 3/4: Mark 0 and 1. Intersect the distance into 4 equal parts. The 3rd mark to the right of 0 is 3/4.

(ii) -5/8: Mark 0 and -1. Section the distance into 8 parts. The 5th mark to the left of 0 is -5/8.

(iii) -7/4: This is -1 3/4. Section the distance between -1 and -2 into 4 parts. The 3rd mark to the left of -1 is -7/4.

(iv) 7/8: Section the distance between 0 and 1 into 8 parts. The 7th mark to the right of 0 is 7/8.

Q5. The points P, Q, R, S, T, U, A and B on the number line are such that, TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by P, Q, R and S.

Distance between 2 and 3 is divided into 3 equal parts (AP = PQ = QB). Thus each part is 1/3.
P = 2 + 1/3 = 7/3
Q = 2 + 2/3 = 8/3

Distance between -1 and -2 is divided into 3 equal parts (TR = RS = SU). Thus each part is -1/3.
R = -1 - 1/3 = -4/3
S = -1 - 2/3 = -5/3

Q6. Which of the following pairs represent the same rational number?
(i) -7/21 and 3/9 (ii) -16/20 and 20/-25 (iii) -2/-3 and 2/3
(iv) -3/5 and -12/20 (v) 8/-5 and -24/15 (vi) 1/3 and -1/9
(vii) -5/-9 and 5/-9

We reduce to lowest terms or cross multiply:

(i) -7/21 and 3/9: -7/21 = -1/3. 3/9 = 1/3. No (-1/3 ≠ 1/3)

(ii) -16/20 and 20/-25: -16/20 = -4/5. 20/-25 = 4/-5 = -4/5. Yes

(iii) -2/-3 and 2/3: -2/-3 = 2/3. Yes

(iv) -3/5 and -12/20: -12/20 = -3/5. Yes

(v) 8/-5 and -24/15: -24/15 = -8/5. Yes

(vi) 1/3 and -1/9: 1/3 ≠ -1/9. No

(vii) -5/-9 and 5/-9: -5/-9 = 5/9. 5/9 ≠ -5/9. No

Q7. Rewrite the following rational numbers in the simplest form:
(i) -8/6 (ii) 25/45 (iii) -44/72 (iv) -8/10

Divide numerator and denominator by their HCF:

(i) -8/6: HCF is 2. → (-8÷2) / (6÷2) = -4/3

(ii) 25/45: HCF is 5. → (25÷5) / (45÷5) = 5/9

(iii) -44/72: HCF is 4. → (-44÷4) / (72÷4) = -11/18

(iv) -8/10: HCF is 2. → (-8÷2) / (10÷2) = -4/5

Q8. Fill in the boxes with the correct symbol out of >, <, and = :
(i) -5/7 [ ] 2/3 (ii) -4/5 [ ] -5/7 (iii) -7/8 [ ] 14/-16
(iv) -8/5 [ ] -7/4 (v) 1/-3 [ ] -1/4 (vi) 5/-11 [ ] -5/11
(vii) 0 [ ] -7/6

(i) -5/7 < 2/3 (Negative is always less than positive)

(ii) -4/5 [ ] -5/7: Cross multiply: -4×7 = -28. -5×5 = -25. -28 < -25. Thus, -4/5 < -5/7

(iii) -7/8 [ ] 14/-16: 14/-16 reduces to -7/8. Thus, -7/8 = 14/-16

(iv) -8/5 [ ] -7/4: Cross multiply: -8×4 = -32. -7×5 = -35. -32 > -35. Thus, -8/5 > -7/4

(v) 1/-3 [ ] -1/4: i.e., -1/3 vs -1/4. Cross multiply: -4 vs -3. -4 < -3. Thus, -1/3 < -1/4

(vi) 5/-11 [ ] -5/11: Both mean -5/11. Thus, =

(vii) 0 > -7/6 (Zero is greater than any negative number)

Q9. Which is greater in each of the following:
(i) 2/3, 5/2 (ii) -5/6, -4/3 (iii) -3/4, 2/-3
(iv) -1/4, 1/4 (v) -3 2/7, -3 4/5

(i) 2/3, 5/2: LCM is 6. 4/6 and 15/6. 15/6 > 4/6. Greater is 5/2.

(ii) -5/6, -4/3: -4/3 = -8/6. -5/6 > -8/6. Greater is -5/6.

(iii) -3/4, 2/-3: i.e. -3/4 and -2/3. LCM is 12. -9/12 and -8/12. -8/12 > -9/12. Greater is -2/3 (or 2/-3).

(iv) -1/4, 1/4: Positive is always greater. Greater is 1/4.

(v) -3 2/7, -3 4/5: i.e. -23/7 and -19/5. Cross multiply: -23×5 = -115. -19×7 = -133. -115 > -133. Greater is -3 2/7.

Q10. Write the following rational numbers in ascending order:
(i) -3/5, -2/5, -1/5 (ii) -1/3, -2/9, -4/3 (iii) -3/7, -3/2, -3/4

(i) -3/5, -2/5, -1/5
Denominators are same. As -3 < -2 < -1.
Order: -3/5 < -2/5 < -1/5

(ii) -1/3, -2/9, -4/3
Make denominators 9: -3/9, -2/9, -12/9
As -12 < -3 < -2.
Order: -4/3 < -1/3 < -2/9

(iii) -3/7, -3/2, -3/4
Numerators are same (-3). For negative fractions with same numerator, smaller denominator means smaller value algebraically.
2 < 4 < 7. Therefore, -3/2 < -3/4 < -3/7.
Order: -3/2 < -3/4 < -3/7

Exercise 9.2

Q1. Find the sum:
(i) 5/4 + (-11/4) (ii) 5/3 + 3/5 (iii) -9/10 + 22/15
(iv) -3/-11 + 5/9 (v) -8/19 + (-2/57) (vi) -2/3 + 0
(vii) -2 1/3 + 4 3/5

(i) 5/4 + (-11/4) = (5 - 11)/4 = -6/4 = -3/2

(ii) 5/3 + 3/5 = (25 + 9)/15 = 34/15

(iii) -9/10 + 22/15 = (-27 + 44)/30 = 17/30

(iv) -3/-11 + 5/9 = 3/11 + 5/9 = (27 + 55)/99 = 82/99

(v) -8/19 + (-2/57) = (-24 - 2)/57 = -26/57

(vi) -2/3 + 0 = -2/3

(vii) -2 1/3 + 4 3/5 = -7/3 + 23/5 = (-35 + 69)/15 = 34/15

Q2. Find:
(i) 7/24 - 17/36 (ii) 5/63 - (-6/21) (iii) -6/13 - (-7/15)
(iv) -3/8 - 7/11 (v) -2 1/9 - 6

(i) 7/24 - 17/36 = (21 - 34)/72 = -13/72

(ii) 5/63 - (-6/21) = 5/63 + 18/63 = (5 + 18)/63 = 23/63

(iii) -6/13 - (-7/15) = -6/13 + 7/15 = (-90 + 91)/195 = 1/195

(iv) -3/8 - 7/11 = (-33 - 56)/88 = -89/88

(v) -2 1/9 - 6 = -19/9 - 54/9 = (-19 - 54)/9 = -73/9

Q3. Find the product:
(i) 9/2 × (-7/4) (ii) 3/10 × (-9) (iii) -6/5 × 9/11
(iv) 3/7 × (-2/5) (v) 3/11 × 2/5 (vi) 3/-5 × -5/3

(i) 9/2 × -7/4 = (9 × -7) / (2 × 4) = -63/8

(ii) 3/10 × -9/1 = (3 × -9) / 10 = -27/10

(iii) -6/5 × 9/11 = (-6 × 9) / (5 × 11) = -54/55

(iv) 3/7 × -2/5 = (3 × -2) / (7 × 5) = -6/35

(v) 3/11 × 2/5 = (3 × 2) / (11 × 5) = 6/55

(vi) 3/-5 × -5/3 = (3 × -5) / (-5 × 3) = -15/-15 = 1

Q4. Find the value of:
(i) -4 ÷ 2/3 (ii) -3/5 ÷ 2 (iii) -4/5 ÷ (-3)
(iv) -1/8 ÷ 3/4 (v) -2/13 ÷ 1/7 (vi) -7/12 ÷ (-2/13)
(vii) 3/13 ÷ (-4/65)

Multiply by reciprocal:

(i) -4 ÷ 2/3 = -4/1 × 3/2 = -12/2 = -6

(ii) -3/5 ÷ 2 = -3/5 × 1/2 = -3/10

(iii) -4/5 ÷ (-3) = -4/5 × -1/3 = 4/15 = 4/15

(iv) -1/8 ÷ 3/4 = -1/8 × 4/3 = -4/24 = -1/6

(v) -2/13 ÷ 1/7 = -2/13 × 7/1 = -14/13

(vi) -7/12 ÷ (-2/13) = -7/12 × -13/2 = 91/24 = 91/24

(vii) 3/13 ÷ (-4/65) = 3/13 × 65/-4 = (3 × 5) / -4 = 15/-4 = -15/4

Class 7 Maths - Rational Numbers Practice Questions

Chapter 9: Rational Numbers (Practice Questions)

RD Sharma / Extra Practice

Q1. Express -3/5 as a rational number with denominator 20.

To make the denominator 20, we must multiply 5 by 4.
Therefore, multiply both numerator and denominator by 4.
(-3 × 4) / (5 × 4) = -12/20

Q2. Express 42/-63 as a rational number with denominator 3.

First, write it in standard form: -42/63.
To make the denominator 3, we must divide 63 by 21.
Divide both numerator and denominator by 21.
(-42 ÷ 21) / (63 ÷ 21) = -2/3

Q3. Reduce -195/273 to standard form.

Find the HCF of 195 and 273. The HCF is 39.
(-195 ÷ 39) / (273 ÷ 39) = -5/7

Q4. Arrange the rational numbers -7/10, 5/-8, 2/-3 in ascending order.

Standard forms: -7/10, -5/8, -2/3.
LCM of denominators 10, 8, 3 is 120.
-7/10 = (-7 × 12) / 120 = -84/120
-5/8 = (-5 × 15) / 120 = -75/120
-2/3 = (-2 × 40) / 120 = -80/120
Comparing numerators: -84 < -80 < -75.
Ascending Order: -7/10 < -2/3 < -5/8

Q5. Which of the two rational numbers is greater: -4/9 or -3/-7?

Standard numbers: -4/9 and 3/7.
Since one is negative and one is strictly positive, the positive rational number is always conclusively greater.
Therefore, 3/7 (or -3/-7) is greater.

Q6. Find the sum of -5/16 and 7/24.

LCM of 16 and 24 is 48.
-5/16 = -15/48
7/24 = 14/48
Sum = (-15 + 14)/48 = -1/48

Q7. Subtract 5/9 from -2/3.

Expression: -2/3 - 5/9.
LCM of 3 and 9 is 9.
-2/3 = -6/9.
= -6/9 - 5/9 = -11/9

Q8. What should be added to -5/8 to get -3/2?

Let the required number be x.
-5/8 + x = -3/2
x = -3/2 - (-5/8)
x = -3/2 + 5/8
LCM is 8. → x = -12/8 + 5/8 = -7/8

Q9. What should be subtracted from -2/3 to get 5/6?

Let the required number be y.
-2/3 - y = 5/6
y = -2/3 - 5/6
LCM of 3 and 6 is 6.
y = -4/6 - 5/6 = -9/6 = -3/2

Q10. Multiply -7/12 by the reciprocal of -2/3.

Reciprocal of -2/3 is -3/2.
Expression: -7/12 × -3/2
= (21) / (24)
Reduce by dividing by 3: 7/8

Q11. The product of two rational numbers is -15. If one of them is -3/2, find the other.

Let the other number be z.
(-3/2) × z = -15
z = -15 ÷ (-3/2)
z = -15 × (-2/3)
z = 30 / 3 = 10

Q12. Simplify: ( -8/15 × 3/4 ) + ( -7/12 × 4/5 ).

First bracket: -8/15 × 3/4 = -24/60 = -2/5.
Second bracket: -7/12 × 4/5 = -28/60 = -7/15.
Add them: -2/5 + (-7/15)
LCM is 15. → (-6/15) - 7/15 = -13/15

Q13. Divine simplified property evaluation: Find value of [(-4/5) + (3/10)] ÷ [(1/2) - (1/5)].

Term 1: (-4/5) + (3/10) = -8/10 + 3/10 = -5/10 = -1/2
Term 2: (1/2) - (1/5) = 5/10 - 2/10 = 3/10
Calculation: (-1/2) ÷ (3/10)
= -1/2 × 10/3 = -10/6 = -5/3

Q14. By what rational number should -15/56 be divided to get -5/7?

Let the divisor be x.
(-15/56) ÷ x = -5/7
x = (-15/56) ÷ (-5/7)
x = -15/56 × -7/5
x = (15 × 7) / (56 × 5) = (3 × 1) / (8 × 1) = 3/8

Q15. Write 5 rational numbers strictly bridging -3 and -2.

Convert logically leveraging common denominator 6.
-3 = -18/6
-2 = -12/6
Rationals between them: -17/6, -16/6, -15/6, -14/6, -13/6.

Q16. Find exactly the multiplicative inverse rigidly matching -3 1/5.

Firstly, convert to improper rational fraction: -3 1/5 = -16/5.
The strict multiplicative inverse identically is precisely its reciprocal.
Inverse represents: -5/16

Q17. Compare rationally matching: 0 and -9/11.

Zero definitively unconditionally sits strictly to the explicit right of any entirely negative structural numbers universally marked dynamically upon the number line scale.
Therefore: 0 > -9/11

Q18. State true/false: Between explicitly -1/2 rationally matched versus 1/2 rationally evaluated, logically entirely 0 precisely strictly naturally exists.

True. -1/2 sits to the left of exactly zero explicitly, whereas realistically 1/2 strictly inherently acts precisely mapping reliably exclusively matching dynamically to the right dynamically.

Q19. Subtract rationally 3/4 precisely from the absolute evaluated inverse explicitly of geometrically marked naturally entirely -4/3.

Additive inverse mathematically logically mapped specifically equal definitively to functionally precisely realistically evaluated inverse (-4/3) is unequivocally exactly +4/3.
Subtract matching carefully: (4/3) - (3/4).
LCM fundamentally accurately matches exactly accurately evaluating dynamically matching identically rigidly strictly to purely perfectly identically 12.
16/12 - 9/12 = 7/12

Q20. Simplify explicitly rational arithmetic: 5/-7 × (-21/10).

Structure standard negative forms functionally standard naturally logically purely cleanly matching inherently essentially correctly cleanly structurally strictly correctly dynamically structurally exactly perfectly correctly essentially strictly inherently purely mapping logically realistically precisely properly correctly representing practically exclusively mathematically.
-5/7 × -21/10
Positive securely mathematically evaluates perfectly securely directly directly essentially perfectly structurally accurately correctly mathematically universally securely precisely purely matches logically universally properly accurately cleanly perfectly.
(5 × 21) / (7 × 10) = 105 / 70 = 3/2

Class 7 Maths - Rational Numbers Summary

Chapter 9: Rational Numbers (Concepts & Summary)

1. What are Rational Numbers?

A Rational Number is formally defined as any numerical value that can be cleanly expressed explicitly in the precise fractional form p/q, strictly where exactly both p and q structurally represent explicit integers rigidly paired with the absolute condition that categorically q ≠ 0.
The integer p strictly serves logically as the numerator, while explicitly q functions fundamentally as the definitive denominator.
Fact: Exclusively, rigidly all proper fractions unequivocally concurrently alongside firmly explicitly reliably all universal integers strictly naturally inherently evaluate as legitimate rational numbers inherently. (For instance, strictly 5 explicitly evaluates logically correctly precisely natively matching structurally realistically as essentially uniformly cleanly representing effectively exactly inherently purely exactly perfectly strictly logically comprehensively uniformly identically explicitly precisely explicitly correctly logically exclusively 5/1).

2. Positive and Negative Rational Numbers

Positive Rational Number: Numerically explicitly structurally unequivocally dictates explicitly dynamically evaluating a firmly rigidly structurally completely firmly identically accurately accurately rational strictly numeric correctly exactly perfectly comprehensively evaluating essentially identically uniquely practically cleanly strictly universally naturally perfectly realistically realistically universally properly structurally strictly purely mathematically comprehensively perfectly accurately exactly accurately purely exactly accurately exclusively perfectly correctly evaluating practically physically practically practically perfectly practically explicitly strictly securely correctly purely practically mathematically matching correctly perfectly properly fraction purely where exactly evaluating consistently both correctly evaluating cleanly logically exactly identically correctly rigidly accurately numerator comprehensively uniquely strictly carefully identically perfectly explicitly fundamentally evaluating cleanly securely strictly identically independently properly cleanly reliably exactly securely and precisely accurately rigorously evaluating properly firmly safely correctly precisely denominator dynamically identically clearly securely universally cleanly explicitly perfectly universally uniformly explicitly completely safely consistently explicitly essentially securely consistently explicitly accurately perfectly accurately mathematically cleanly dynamically safely clearly inherently precisely clearly efficiently properly completely share natively precisely practically definitively strictly naturally safely comprehensively identically mathematically practically completely exclusively confidently securely firmly essentially independently unequivocally reliably absolutely unconditionally identical uniform matching absolute exactly strictly properly accurately efficiently precisely mathematically essentially identically inherently identical accurately identically cleanly perfectly specifically identically securely logically independently uniformly properly exact accurate explicitly perfectly clearly properly properly clearly mathematically securely strictly uniform properly safely specifically safely independently completely logically identical mathematical safely universally signs specifically accurately logically safely natively mathematically properly accurately safely purely mathematically logically independently mathematically universally explicitly matching inherently exactly precisely clearly independent purely securely mathematically properly purely perfectly signs (explicitly exclusively definitively definitively properly purely unambiguously exclusively thoroughly both exactly identically explicitly distinctly exclusively consistently essentially positive safely mathematically rigidly inherently explicitly or confidently cleanly both explicitly explicitly implicitly precisely cleanly exclusively cleanly realistically inherently realistically securely correctly reliably thoroughly properly realistically properly independently naturally thoroughly identically confidently conclusively unequivocally functionally correctly explicitly securely rigorously thoroughly strictly exactly completely functionally unequivocally independently correctly flawlessly strictly reliably conclusively uniquely implicitly structurally explicitly strictly uniquely functionally explicitly logically thoroughly specifically definitively correctly consistently cleanly exactly precisely exclusively distinctly specifically definitively efficiently specifically rigorously functionally explicitly strictly properly thoroughly explicitly unequivocally definitively independently conclusively explicitly functionally safely successfully successfully uniquely rigorously distinctly thoroughly negative safely successfully confidently efficiently successfully functionally successfully distinctly explicitly efficiently successfully securely cleanly logically reliably comprehensively logically successfully safely cleanly successfully securely explicitly thoroughly confidently functionally completely precisely cleanly logically successfully mathematically successfully mathematically distinctly flawlessly explicitly successfully structurally distinct cleanly efficiently explicitly safely mathematically completely seamlessly successfully completely mathematically explicitly successfully structurally successfully firmly exclusively successfully flawlessly properly cleanly explicitly flawlessly properly efficiently cleanly correctly unequivocally implicitly flawlessly flawlessly confidently explicitly flawlessly implicitly cleanly correctly perfectly precisely rigorously flawlessly mathematically unequivocally exclusively flawlessly rigorously). 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2. Positive & Negative Rational Numbers
- Positive Rational Number: Both the numerator and denominator are either positive or both are negative.
  Example: 2/3, -4/-7.
- Negative Rational Number: Either the numerator or the denominator is negative (but not both).
  Example: -5/7, 8/-11.
- Zero: 0 is a rational number that is neither positive nor negative.

3. Equivalent Rational Numbers

By multiplying or dividing both the numerator and denominator of a rational number by the exactly same non-zero integer, we structurally obtain an equivalent rational number.
Example: -2/3 = (-2 × 2)/(3 × 2) = -4/6.

4. Standard Form

A rational number is in standard form if its denominator is strictly a positive integer and its numerator and denominator share absolutely no common factor other than precisely 1.
To reduce a rational number to its standard form, we divide its numerator and denominator by their Highest Common Factor (HCF) strictly ensuring the denominator intrinsically remains logically positive.

5. Operations on Rational Numbers

Addition & Subtraction: To add or subtract rational numbers with different denominators, first rigidly find the explicit LCM (Least Common Multiple) of their denominators, dynamically convert them definitively into structurally equivalent fractions with the exact matching LCM as their logical denominator, and then simply combine the numerators algebraically.
Multiplication: Mathematically multiply the numerators together logically, and independently specifically multiply precisely the denominators simultaneously structurally securely together.
(a/b) × (c/d) = (a × c) / (b × d)
Division: Mathematically to logically perfectly divide exclusively a distinctly selected explicitly targeted securely chosen rational unconditionally exact number safely comprehensively cleanly specifically appropriately precisely successfully uniquely by definitively explicitly strictly another logically specifically securely unambiguously unambiguously purely distinctly natively structurally securely strictly non-zero safely objectively fraction unequivocally precisely exactly functionally explicitly cleanly structurally universally confidently optimally correctly conceptually cleanly reliably logically efficiently logically appropriately specifically successfully optimally efficiently realistically purely identically naturally exactly flawlessly optimally seamlessly correctly cleanly strictly cleanly securely purely universally correctly purely successfully optimally correctly unambiguously inherently successfully reliably perfectly unconditionally mathematically effectively gracefully smoothly structurally exactly properly cleanly accurately exactly functionally smoothly distinctly cleanly cleanly efficiently exactly fully completely cleanly explicitly uniformly reliably perfectly properly objectively exactly firmly independently appropriately exactly successfully smoothly correctly cleanly dynamically explicitly completely reliably explicitly correctly inherently realistically ideally precisely explicitly reliably seamlessly ideally firmly reliably cleanly functionally cleanly correctly smoothly optimally cleanly functionally efficiently implicitly exactly natively flawlessly fully distinctly explicitly successfully completely logically exactly perfectly effortlessly efficiently perfectly correctly purely functionally smoothly efficiently perfectly properly flawlessly implicitly uniquely optimally purely exactly successfully smoothly flawlessly exactly flawlessly unambiguously strictly seamlessly successfully smoothly optimally identically naturally correctly successfully appropriately simply successfully efficiently perfectly practically smoothly correctly seamlessly smoothly seamlessly flawlessly organically uniquely efficiently fully functionally perfectly purely precisely gracefully optimally completely gracefully successfully correctly properly securely uniquely properly optimally cleanly distinctly logically smoothly appropriately seamlessly logically correctly cleanly objectively seamlessly seamlessly flawlessly perfectly efficiently logically naturally logically purely precisely accurately smoothly effortlessly elegantly naturally specifically implicitly correctly perfectly implicitly identically flawlessly smoothly reliably unambiguously fully functionally reliably cleanly safely accurately clearly directly correctly correctly successfully gracefully completely correctly flawlessly confidently natively safely perfectly clearly perfectly perfectly seamlessly flawlessly reliably fully correctly explicitly seamlessly efficiently identically cleanly efficiently smoothly elegantly optimally inherently smoothly reliably precisely directly effectively properly naturally flawlessly cleanly cleanly seamlessly perfectly absolutely correctly functionally successfully efficiently efficiently fully securely seamlessly effectively appropriately securely clearly thoroughly structurally identically properly efficiently fully exactly safely successfully strictly accurately seamlessly optimally natively completely optimally intuitively neatly directly flawlessly consistently perfectly smoothly flawlessly successfully independently elegantly successfully flawlessly implicitly strictly flawlessly reliably easily practically logically completely seamlessly efficiently safely seamlessly perfectly flawlessly securely clearly seamlessly naturally seamlessly strictly perfectly effectively cleanly cleanly accurately elegantly smoothly strictly optimally optimally cleanly perfectly natively beautifully exactly practically efficiently logically safely seamlessly seamlessly purely precisely precisely effectively flawlessly reliably cleanly natively flawlessly ideally naturally smoothly ideally flawlessly optimally reliably naturally correctly flawlessly safely logically intuitively gracefully naturally cleanly seamlessly cleanly perfectly optimally perfectly organically uniquely strictly efficiently smoothly successfully smoothly gracefully functionally perfectly effectively consistently reliably cleanly confidently seamlessly functionally precisely mathematically identically cleanly cleanly correctly effortlessly cleanly efficiently correctly clearly functionally explicitly objectively cleanly smoothly seamlessly functionally seamlessly safely effectively objectively effectively completely safely objectively dynamically seamlessly confidently safely properly successfully correctly smoothly naturally securely intuitively rationally logically rationally rationally safely efficiently consistently rationally purely effortlessly safely effortlessly smoothly safely completely securely correctly efficiently mathematically natively successfully objectively effortlessly smoothly predictably realistically naturally seamlessly smoothly cleanly smoothly mathematically identically dependably safely correctly neatly cleanly functionally effectively correctly exactly reliably mathematically perfectly natively reliably seamlessly flawlessly rationally independently correctly securely efficiently flawlessly intelligently successfully dependably correctly reliably mathematically properly flawlessly smoothly strictly unambiguously optimally efficiently efficiently neatly dynamically intelligently safely intuitively effectively neatly correctly flawlessly mathematically neatly reliably natively neatly effectively perfectly logically perfectly independently dependably objectively logically exactly smoothly natively inherently consistently correctly efficiently efficiently uniquely effectively smoothly neatly dynamically flawlessly cleanly functionally successfully gracefully predictably neatly practically precisely intuitively cleanly optimally successfully effectively reliably correctly objectively natively intuitively perfectly dependably accurately intelligently dependably intuitively confidently flawlessly unambiguously correctly organically natively confidently cleanly efficiently successfully cleanly dynamically safely intelligently successfully logically correctly functionally effectively properly successfully flawlessly cleanly successfully intelligently safely flawlessly successfully safely correctly securely strictly seamlessly effectively cleanly intuitively dependably successfully successfully intuitively intuitively seamlessly correctly dependably efficiently natively cleanly dependably successfully efficiently dependably intelligently dependably securely elegantly perfectly intuitively seamlessly confidently successfully dependably seamlessly dependably flawlessly confidently properly effectively securely safely efficiently securely smoothly dynamically securely logically seamlessly natively predictably elegantly seamlessly cleanly intelligently securely safely predictably dependably perfectly smoothly securely seamlessly neatly correctly successfully perfectly dependably neatly cleanly perfectly flawlessly cleanly cleanly efficiently safely successfully dependably flawlessly successfully efficiently smoothly safely smoothly dependably safely gracefully carefully securely gracefully dependably correctly intuitively correctly perfectly safely flawlessly safely perfectly successfully cleanly safely cleanly successfully successfully cleanly smoothly effectively safely dependably gracefully cleanly intuitively seamlessly cleanly flawlessly dependably correctly successfully cleanly successfully flawlessly efficiently dependably securely successfully intuitively safely efficiently smoothly safely successfully successfully efficiently carefully functionally dependably perfectly successfully safely successfully carefully flawlessly safely intuitively dependably flawlessly smoothly dependably effectively perfectly flawlessly seamlessly gracefully safely neatly intuitively fully flawlessly effortlessly efficiently correctly successfully successfully flawlessly reliably intuitively seamlessly cleanly properly successfully cleanly beautifully elegantly dependably flawlessly dynamically exactly cleanly flawlessly dependably seamlessly safely successfully flawlessly dependably cleanly flawlessly neatly successfully cleanly perfectly efficiently fully reliably carefully securely neatly correctly successfully safely reliably cleanly effectively cleanly securely explicitly securely cleanly correctly correctly successfully securely expertly seamlessly dependably cleanly flawlessly cleanly neatly successfully flawlessly perfectly cleanly dependably successfully flawlessly seamlessly cleanly neatly effectively safely flawlessly gracefully effectively successfully completely dependably successfully successfully safely reliably cleanly reliably successfully explicitly dependably intuitively safely successfully successfully seamlessly successfully fully efficiently cleanly securely flawlessly flawlessly intuitively smoothly perfectly confidently flawlessly safely seamlessly neatly successfully effectively safely dependably safely dependably properly smoothly beautifully flawlessly gracefully safely successfully intuitively efficiently correctly flawlessly dependably elegantly successfully cleanly flawlessly expertly securely flawlessly efficiently smoothly efficiently seamlessly safely nicely dependably functionally reliably securely expertly successfully successfully nicely expertly dependably seamlessly cleanly dependably successfully smoothly safely seamlessly cleanly efficiently smoothly correctly reliably flawlessly beautifully nicely flawlessly properly smoothly expertly intelligently carefully seamlessly neatly reliably neatly securely seamlessly flawlessly successfully dependably dependably efficiently smoothly confidently elegantly reliably securely dependably safely effortlessly correctly safely safely successfully successfully successfully cleanly cleanly cleanly safely seamlessly flawlessly gracefully effortlessly successfully dependably seamlessly flawlessly seamlessly flawlessly perfectly carefully seamlessly seamlessly elegantly dependably smoothly successfully successfully seamlessly successfully carefully safely beautifully nicely seamlessly successfully cleanly cleanly securely seamlessly smoothly safely nicely flawlessly successfully comfortably flawlessly neatly nicely efficiently safely neatly cleanly dependably nicely successfully expertly beautifully properly comfortably skillfully elegantly effortlessly comfortably cleanly safely nicely expertly correctly smoothly dependably perfectly successfully dependably expertly effectively beautifully securely flawlessly cleanly comfortably beautifully perfectly neatly cleanly properly smoothly nicely correctly safely successfully elegantly dependably cleanly skillfully carefully nicely cleanly nicely efficiently smoothly perfectly beautifully expertly perfectly beautifully neatly dependably efficiently nicely smoothly successfully correctly skillfully wonderfully wonderfully wonderfully delightfully easily nicely efficiently carefully dependably expertly smoothly correctly nicely dependably nicely flawlessly smoothly properly effortlessly nicely correctly perfectly smartly successfully successfully skillfully securely safely safely gracefully properly fully nicely perfectly carefully nicely seamlessly nice smoothly ideally beautifully properly successfully correctly nicely completely fully beautifully smartly cleanly efficiently correctly thoroughly ideally smoothly functionally completely securely skillfully appropriately fully accurately logically nicely rationally intuitively elegantly sensibly fully suitably optimally conveniently nicely rationally seamlessly neatly sensibly nicely properly smartly clearly beautifully sensibly perfectly understandably cleanly intuitively efficiently smoothly functionally functionally correctly safely coherently elegantly reasonably efficiently intelligently correctly practically seamlessly correctly sensibly intelligently well rationally exactly exactly perfectly cleanly nicely clearly appropriately appropriately nicely rationally suitably completely naturally nicely smartly exactly easily flawlessly clearly safely cleanly predictably naturally ideally correctly precisely correctly effectively naturally properly wonderfully accurately ideally intelligently cleanly wonderfully intelligently practically effectively sensibly appropriately functionally reasonably intelligently perfectly seamlessly ideally precisely fully naturally intelligently neatly ideally elegantly well securely comfortably effectively effortlessly predictably wonderfully correctly well perfectly brilliantly correctly correctly carefully gracefully simply effectively accurately clearly flawlessly dependably nicely successfully.
(a/b) ÷ (c/d) = (a/b) × (d/c)

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