Decimal and Fractions: AFCAT Numerical Ability

Formulas (3)

• Reading and writing decimals in words · A decimal with n places
  $$\text{decimal} = \dfrac{\text{the digits after the point}}{10^n} \quad (n = \text{number of places})$$
• Writing a fraction as a decimal · Fraction to decimal
  $$\dfrac ab = \dfrac{a \times k}{b \times k} \quad \text{with } b \times k = 10, 100, 1000, \dots$$
• Writing a decimal as a fraction in lowest terms · Decimal to fraction
  $$0.d_1 d_2 \dots d_n = \dfrac{d_1 d_2 \dots d_n}{10^n}, \text{ then cancel the HCF}$$

Reference tables (1)

Watch out for (7)

• A zero holds a place
  → Place value after the decimal point
• Places are counted from the point
  → Place value after the decimal point
• Count places, not digits
  → Reading and writing decimals in words
• The digits of the fraction are not the decimal
  → Writing a fraction as a decimal
• Eighths need three places
  → Writing a fraction as a decimal
• Lowest terms means no common factor left
  → Writing a decimal as a fraction in lowest terms
• One zero in the denominator per place
  → Writing a decimal as a fraction in lowest terms

Formulas (3)

• Terminating or recurring: the denominator test · The terminating test
  $$\dfrac ab \text{ in lowest terms terminates} \;\text{means}\; b = 2^m \times 5^n$$
• Pure recurring decimals as fractions · Pure recurring decimal
  $$0.\overline{d_1 d_2 \dots d_n} = \dfrac{d_1 d_2 \dots d_n}{\text{as many 9s as there are repeating digits}}$$
• Mixed recurring decimals as fractions · Mixed recurring decimal
  $$0.a\overline{b} = \dfrac{ab - a}{90}, \quad 0.a\overline{bc} = \dfrac{abc - a}{990}, \quad 0.ab\overline{c} = \dfrac{abc - ab}{900}$$

Watch out for (6)

• Reduce before you test
  → Terminating or recurring: the denominator test
• A rounded decimal is a different number
  → Terminating or recurring: the denominator test
• Nines, not a power of ten
  → Pure recurring decimals as fractions
• The whole part is added, not written on top
  → Pure recurring decimals as fractions
• Subtract the part that does not repeat
  → Mixed recurring decimals as fractions
• One 0 for each digit that does not repeat
  → Mixed recurring decimals as fractions

Formulas (4)

• Comparing two decimals · Padding with zeros
  $$0.7 = 0.70 = 0.700$$
• Ordering a list, and picking the largest or smallest · Order by padded value
  $$0.5,\ 0.505,\ 0.05 \;\to\; 500,\ 505,\ 50 \text{ thousandths}$$
• A decimal between two decimals · Between two decimals
  $$a < x < b$$
• Comparing and ordering fractions · Cross-multiplying
  $$\dfrac ab > \dfrac cd \;\text{means}\; ad > bc$$

Watch out for (7)

• More digits is not bigger
  → Comparing two decimals
• Trailing zeros change nothing
  → Comparing two decimals
• Read which order is asked
  → Ordering a list, and picking the largest or smallest
• A longer decimal can be smaller
  → Ordering a list, and picking the largest or smallest
• Check against both ends
  → A decimal between two decimals
• A bigger denominator is smaller only when the numerators match
  → Comparing and ordering fractions
• Cross-multiply the right pairs
  → Comparing and ordering fractions

Formulas (3)

• Adding and subtracting decimals · Line up the points
  $$6.07 + 0.935 = 6.070 + 0.935$$
• Multiplying decimals · Places in a product
  $$\text{places in } a \times b = \text{places in } a + \text{places in } b$$
• Dividing decimals · Clearing the divisor
  $$\dfrac ab = \dfrac{a \times 10^n}{b \times 10^n}$$

Watch out for (6)

• Line up the points, not the last digits
  → Adding and subtracting decimals
• A whole number still has places
  → Adding and subtracting decimals
• Place the point before dropping zeros
  → Multiplying decimals
• Multiplying by a decimal below 1 makes it smaller
  → Multiplying decimals
• Move the point in both numbers
  → Dividing decimals
• Count the zeros in a small answer
  → Dividing decimals

Formulas (6)

• Operations on fractions and mixed numbers · Dividing by a fraction
  $$\dfrac ab \div \dfrac cd = \dfrac ab \times \dfrac dc$$
• Finding the whole from a fraction of it · Whole from a part
  $$\text{whole} = \dfrac{\text{amount}}{\text{the fraction it is (or the difference of two fractions)}}$$
• The wrong-operation problem · Wrong result against right result
  $$\left(\dfrac{q}{p} - \dfrac{p}{q}\right) x = d \quad (\text{divided by } \tfrac pq \text{ instead of multiplying})$$
• Order of operations (BODMAS) · Order of operations
  $$\text{brackets} \to \text{of} \to \div \text{ and } \times \to + \text{ and } -$$
• Complex and continued fractions · A fraction over a fraction
  $$\dfrac{a / b}{c / d} = \dfrac ab \times \dfrac dc$$
• Algebraic shortcuts in simplification · Difference of squares
  $$a^2 - b^2 = (a - b)(a + b)$$

Watch out for (12)

• Mixed numbers multiply only as improper fractions
  → Operations on fractions and mixed numbers
• Of means times
  → Operations on fractions and mixed numbers
• Divide by the difference, not by either fraction
  → Finding the whole from a fraction of it
• Remains means the new level
  → Finding the whole from a fraction of it
• Dividing by a fraction flips it
  → The wrong-operation problem
• Answer what is asked
  → The wrong-operation problem
• Division and multiplication go left to right
  → Order of operations (BODMAS)
• A minus before a bracket flips the signs inside
  → Order of operations (BODMAS)
• Never cancel across a plus sign
  → Complex and continued fractions
• A continued fraction is read from the bottom
  → Complex and continued fractions
• The zero needs a + b + c = 0
  → Algebraic shortcuts in simplification
• Look for the shape before multiplying
  → Algebraic shortcuts in simplification

Formulas (3)

• Perfect squares and cubes, and the nearest square · Reaching the nearest square
  $$n^2 \le N < (n + 1)^2: \quad \text{add } (n + 1)^2 - N \text{ or subtract } N - n^2$$
• Square and cube roots of decimals · Roots of decimals
  $$\sqrt{\dfrac{a}{10^{2n}}} = \dfrac{\sqrt a}{10^n}, \qquad \sqrt[3]{\dfrac{a}{10^{3n}}} = \dfrac{\sqrt[3]{a}}{10^n}$$
• Nested and never-ending roots · A never-ending root
  $$x = \sqrt{a + \sqrt{a + \dots}} \;\Rightarrow\; x^2 = a + x$$

Watch out for (6)

• Add uses the square above, subtract the square below
  → Perfect squares and cubes, and the nearest square
• Find the range from the number of digits
  → Perfect squares and cubes, and the nearest square
• An odd number of places has no neat root
  → Square and cube roots of decimals
• Halve the places, do not keep them
  → Square and cube roots of decimals
• Start at the innermost root
  → Nested and never-ending roots
• Keep the positive root
  → Nested and never-ending roots

Formulas (4)

• Decimals to percentages and back · Decimal and percentage
  $$x = (100x)\%, \qquad p\% = \dfrac{p}{100}$$
• A percentage of a number · Percent of a number
  $$p\% \text{ of } N = \dfrac{p}{100} \times N$$
• What percentage one number is of another · Part as a percentage of the whole
  $$\dfrac{\text{part}}{\text{whole}} \times 100\%$$
• Percentage increase and decrease · Percentage change
  $$\text{change}\% = \dfrac{\text{new} - \text{original}}{\text{original}} \times 100$$

Watch out for (6)

• Two places, not one
  → Decimals to percentages and back
• Going back divides
  → Decimals to percentages and back
• Percent of is not percent off
  → A percentage of a number
• The whole goes underneath
  → What percentage one number is of another
• Divide by the original, not the new value
  → Percentage increase and decrease
• Up and down by the same percentage does not cancel
  → Percentage increase and decrease