Decimals and Fractions Concepts

1. Fraction to Decimal Conversion

• Formula: $\text{Decimal}=\frac{\text{Numerator}}{\text{Denominator}}$

• Example: Question: Convert (\frac{3}{8}) into decimal form.

  Solution:

  $\frac38=0.375$

  Answer: (\boxed{0.375})

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2. Decimal to Fraction Conversion

• Formula:

  For a terminating decimal, remove the decimal point and divide by the corresponding power of 10.

  $0.375=\frac{375}{1000}=\frac38$

• Example: Question: Convert (0.625) into a fraction.

  Solution:

  $0.625=\frac{625}{1000}$

  Divide by 125:

  $=\boxed{\frac58}$

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3. Percentage, Fraction and Decimal Conversion

• Formula:

  $\text{Percentage}=\text{Decimal}\times100$

  $\text{Decimal}=\frac{\text{Percentage}}{100}$

  $\text{Fraction}=\frac{\text{Percentage}}{100}$

• Example: Question: Express (\frac38) as a decimal and percentage.

  Solution:

  $\frac38=0.375$

  $0.375\times100=37.5$

  Answer:

  \boxed{0.375=37.5%}

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4. Recurring Decimal to Fraction — Pure Recurring

• Formula:

  $0.\overline{a}=\frac{a}{9}$

  $0.\overline{ab}=\frac{ab}{99}$

  $0.\overline{abc}=\frac{abc}{999}$

• Example: Question: Convert (0.\overline{27}) into a fraction.

  Solution:

  $0.\overline{27}=\frac{27}{99}$

  $=\boxed{\frac3{11}}$

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5. Recurring Decimal to Fraction — Mixed Recurring

• Formula:

  \frac{\text{Number formed by all digits}-\text{non-recurring part}} {\text{9s for recurring digits followed by 0s for non-recurring digits}}$$

• Example: Question: Convert (0.2\overline{36}) into a fraction.

  Solution:

  $0.2\overline{36}=\frac{236-2}{990}$

  $=\frac{234}{990}$

  $=\boxed{\frac{13}{55}}$

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6. Comparing Two Fractions

• Formula:

  For positive denominators:

  $\frac ab>\frac cd\iff ad>bc$

• Example: Question: Which is greater: (\frac{7}{12}) or (\frac58)?

  Solution:

  Cross multiply:

  $7\times8=56$

  $5\times12=60$

  Since (60>56),

  $\boxed{\frac58>\frac7{12}}$

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7. Ordering Fractions

• Example: Question: Arrange (\frac34,\frac57,\frac7{10}) in ascending order.

  Solution:

  Compare:

  $\frac34\quad\text{and}\quad\frac57$

  $3\times7=21,\qquad5\times4=20$

  Therefore,

  $\frac34>\frac57$

  Next:

  $\frac57\quad\text{and}\quad\frac7{10}$

  $5\times10=50,\qquad7\times7=49$

  Therefore,

  $\frac57>\frac7{10}$

  Hence:

  $\boxed{\frac7{10}<\frac57<\frac34}$

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8. Simplifying Fractions

• Formula:

  Divide numerator and denominator by their HCF.

• Example: Question: Simplify (\frac{84}{126}).

  Solution:

  HCF of 84 and 126 is 42.

  $\frac{84}{126}=\frac{84\div42}{126\div42}$

  $=\boxed{\frac23}$

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9. Addition and Subtraction of Fractions

• Formula:

  $\frac ab+\frac cd=\frac{ad+bc}{bd}$

  $\frac ab-\frac cd=\frac{ad-bc}{bd}$

• Example: Question: Find:

  $\frac23+\frac34$

  Solution:

  =\frac{2(4)+3(3)}{12}$$ $$=\frac{8+9}{12}$$ $$=\boxed{\frac{17}{12}}$$

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10. Multiplication and Division of Fractions

• Formula:

  $\frac ab\times\frac cd=\frac{ac}{bd}$

  $\frac ab\div\frac cd=\frac ab\times\frac dc$

• Example: Question: Find:

  $\frac35\div\frac9{10}$

  Solution:

  $\frac35\times\frac{10}{9}$

  $=\frac{30}{45}$

  $=\boxed{\frac23}$

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11. Fraction of a Number

• Formula:

  $\text{Fraction of number}=\text{Number}\times\text{Fraction}$

• Example: Question: Find (\frac35) of 250.

  Solution:

  $250\times\frac35$

  $=50\times3$

  $=\boxed{150}$

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12. Finding the Whole From a Fraction

• Example: Question: (\frac35) of a number is 48. Find the number.

  Solution:

  Let the number be (x).

  $\frac35x=48$

  $x=48\times\frac53$

  $=\boxed{80}$

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13. Terminating vs Recurring Decimals

• Rule:

  A fraction in lowest form has a terminating decimal only when its denominator contains no prime factors other than 2 and/or 5.

• Example: Question: Which of (\frac38,\frac7{20},\frac5{12}) has a terminating decimal?

  Solution:

  $8=2^3$

  So (\frac38) terminates.

  $20=2^2\times5$

  So (\frac7{20}) terminates.

  $12=2^2\times3$

  Since 3 is present, (\frac5{12}) is recurring.

  Answer:

  $\boxed{\frac38,\frac7{20}}$

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14. Number of Decimal Places

• Example: Question: How many decimal places are needed to express (\frac7{40}) as a terminating decimal?

  Solution:

  $\frac7{40}=\frac{175}{1000}$

  Therefore:

  $\boxed{0.175}$

  It has 3 decimal places.

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15. Recurring Decimal Comparison

• Example: Question: Which is greater: (0.\overline6) or (0.65)?

  Solution:

  $0.\overline6=\frac23=0.666\ldots$

  Therefore:

  $\boxed{0.\overline6>0.65}$

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16. Mixed Decimal Comparison

• Example: Question: Which is greater: (0.72) or (0.7\overline1)?

  Solution:

  $0.7\overline1=0.71111\ldots$

  Compare:

  $0.72000\ldots>0.71111\ldots$

  Therefore:

  $\boxed{0.72>0.7\overline1}$

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17. Approximation of Numbers

• Rule:

  To round to (n) decimal places, inspect the next digit.

  • (0)–(4): keep the digit.
  • (5)–(9): increase the digit by 1.

• Example: Question: Round (8.3764) to 2 decimal places.

  Solution:

  The second decimal digit is (7).

  The next digit is (6), so round (7) up to (8).

  $\boxed{8.38}$

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18. Approximation in Multiplication

• Example: Question: Estimate (49.8\times20.2).

  Solution:

  Approximate:

  $49.8\approx50$

  $20.2\approx20$

  Therefore:

  $50\times20=\boxed{1000}$

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19. Comparing Fractions Using a Common Reference

• Example: Question: Which is closest to 1: (\frac{9}{10},\frac{11}{12},\frac{14}{15})?

  Solution:

  Find how far each is from 1:

  $1-\frac9{10}=\frac1{10}$

  $1-\frac{11}{12}=\frac1{12}$

  $1-\frac{14}{15}=\frac1{15}$

  The smallest difference is (\frac1{15}).

  Therefore:

  $\boxed{\frac{14}{15}}$

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Advanced Variants

20. Recurring Decimal With More Than One Non-Recurring Digit

• Example: Question: Convert (0.12\overline{34}) into a fraction.

  Solution:

  =\frac{1234-12}{9900}$$ $$=\frac{1222}{9900}$$ $$=\boxed{\frac{611}{4950}}$$

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21. Recurring Decimal in an Equation

• Example: Question: Solve:

  $x+0.\overline3=1$

  Solution:

  $0.\overline3=\frac13$

  Therefore:

  $x+\frac13=1$

  $x=\frac23$

  Answer:

  $\boxed{\frac23}$

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22. Fractional Expression With Nested Fractions

• Example: Question: Find:

  $\frac{1}{1+\frac12}$

  Solution:

  First simplify the denominator:

  $1+\frac12=\frac32$

  Therefore:

  $\frac1{3/2}=\frac23$

  Answer:

  $\boxed{\frac23}$

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23. Fractional Expression With Multiple Operations

• Example: Question: Find:

  $\frac12+\frac34-\frac18$

  Solution:

  Take denominator (8):

  $\frac48+\frac68-\frac18$

  $=\frac98$

  Answer:

  $\boxed{\frac98}$

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24. Recurring Decimal Difference

• Example: Question: Find:

  $0.\overline{27}-0.\overline{18}$

  Solution:

  $0.\overline{27}=\frac3{11}$

  $0.\overline{18}=\frac2{11}$

  Therefore:

  =\boxed{\frac1{11}}$$

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