a/b>1/2 if 2a>b.
a/b=1/2 if 2a=b.
16. Fractions on the Number Line
Remember:
Positive proper fractions lie between0 and1.
Equivalent fractions occupy the same point on the number line.
17. Rational Numbers Between Fractions
Property:
Infinitely many rational numbers exist between any two distinct rational numbers.
18. Addition of Like Fractions
Formula:
a/c+b/c=(a+b)/c
19. Addition of Unlike Fractions
Method:
Convert the fractions to a common denominator before adding.
Shortcut:
Use the LCM of denominators rather than their product whenever possible.
20. Subtraction of Fractions
Rule:
Like addition, unlike fractions must first be converted to a common denominator before subtraction.
21. Multiplication of Fractions
Formula:
a/b×c/d=ac/bd
Shortcut:
Cancel common factors before multiplication.
22. Reciprocal of a Fraction
Rule:
Reciprocal of a/b is b/a, provided a≠0.
Important:
Zero has no reciprocal.
23. Division of Fractions
Formula:
a/b÷c/d=a/b×d/c
Memory Rule:
Keep the first fraction, change division to multiplication and invert the divisor.
24. Meaning of “Of”
Rule:
In ordinary aptitude questions, “of” means multiplication.
3/5 of40=3/5×40=24.
25. Fraction Operations and BODMAS
Order:
Brackets → Orders → Division/Multiplication → Addition/Subtraction
Note:
Division and multiplication have equal priority and are performed left to right.
26. Cancellation Rule
Shortcut:
During multiplication, common factors of numerators and denominators may be cancelled.
Exam Trap:
Do not cancel terms separated by + or −.
27. Decimal Place Value
Right of Decimal Point:
1st place → Tenths
2nd place → Hundredths
3rd place → Thousandths
4th place → Ten-thousandths
28. Face Value and Place Value
Face Value: Digit itself.
Place Value: Digit × value of its position.
29. Equivalent Decimals
Rule:
Trailing zeros do not change the value of a terminating decimal.
4.5=4.50=4.500
Remember:
4.05 is not equal to4.5.
30. Comparing Decimals
Method:
Compare whole-number parts first, then tenths, hundredths, thousandths and so on.
Shortcut:
Temporarily add trailing zeros to equalise decimal places.
31. Addition and Subtraction of Decimals
Key Rule:
Align decimal points vertically before adding or subtracting.
32. Multiplication of Decimals
Method:
Ignore decimal points temporarily, multiply as whole numbers and then place the decimal according to the total decimal places in the factors.
33. Division by a Decimal
Rule:
Make the divisor a whole number by multiplying both dividend and divisor by the same power of10.
34. Multiplication by Powers of 10
Rule:
×10 → Decimal1 place right
×100 →2 places right
×1000 →3 places right
35. Division by Powers of 10
Rule:
÷10 → Decimal1 place left
÷100 →2 places left
÷1000 →3 places left
36. Fraction to Decimal Conversion
Rule:
a/b=a÷b.
Example:
7/8=0.875.
37. Terminating Decimal to Fraction
Method:
Remove the decimal point, use10n as denominator where n is the number of decimal places, then simplify.
Example:
0.375=375/1000=3/8.
38. Terminating Decimal Criterion
Fundamental Rule:
After reducing p/q to lowest terms, the decimal terminates if and only if q has no prime factors other than2 and5.
39. Non-Terminating Recurring Criterion
Rule:
If the reduced denominator contains any prime factor other than2 or5, its decimal expansion is non-terminating recurring.
40. Number of Decimal Places
Rule:
If the reduced denominator is2m5n, the terminating decimal has at most:
max(m,n) decimal places.
41. Pure Recurring Decimal — One Digit
Rule:
0.aaa...=a/9.
Example:
0.777...=7/9.
42. Pure Recurring Decimal — Two Digits
Rule:
0.ababab...=ab/99.
Example:
0.272727...=27/99=3/11.
43. Pure Recurring Decimal — Three Digits
Rule:
Three repeating digits use denominator999.
0.125125...=125/999.
44. Mixed Recurring Decimal
Shortcut:
Numerator = Number formed up to one complete recurring block − Number formed by the non-recurring part.
Denominator =9s for recurring digits followed by0s for non-recurring decimal digits.
45. Rational Decimal Expansions
Property:
Every rational number has either a terminating or a non-terminating recurring decimal expansion.
46. Non-Terminating Non-Recurring Decimals
Property:
A non-terminating non-recurring decimal represents an irrational number.
Exam Tip:
Non-terminating does not automatically mean irrational; recurring decimals are rational.
47. Choosing Fraction or Decimal Form
Exam Strategy:
Use decimals when conversions are simple and addition/subtraction dominates.
Use fractions when cancellation, exact calculation or recurring decimals are involved.
48. Remaining-Part Problems
Rule:
If fraction a of the whole is used:
Remaining fraction=1−a.
49. Successive Remaining-Part Problems
Rule:
If fraction a is used first and then fraction b of the remainder is used:
Final remaining fraction=(1−a)(1−b).
Exam Trap:
A fraction of the remainder is not the same as the same fraction of the original quantity.
50. Final Chapter Strategy
Before Solving:
• Identify whether fraction or decimal form is easier.
• Simplify fractions early.
• Use cancellation before multiplication.
• Align decimal points in addition/subtraction.
• Follow BODMAS.
• Keep recurring decimals exact when possible.
• Check whether a fraction is of the whole or of the remainder.
• Ensure units are compatible.
• Estimate the answer to detect decimal-point errors.