Basic Arithmetic Operations
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Basic Arithmetic Operations — Quick Facts
MAT tests your speed and accuracy with the four fundamental operations. Every question ultimately relies on your comfort with addition, subtraction, multiplication, and division of integers, fractions, and decimals.
Key Shortcuts for MAT:
- Multiplication shortcuts: When multiplying by 5, multiply by 10 then halve. When multiplying by 25, multiply by 100 then divide by 4. When multiplying by 125, multiply by 1000 then divide by 8.
- Division shortcuts: To divide by 5, multiply by 2 then divide by 10. To divide by 25, multiply by 4 then divide by 100. To divide by 125, multiply by 8 then divide by 1000.
- Squaring shortcuts: For numbers ending in 5, use $n \times (n+1)$ appending 25. For example, $65^2 = 6 \times 7 = 42$, append 25 → 4225. For numbers near 50, use $(50 \pm d)^2 = 2500 \pm 100d + d^2$.
BODMAS Rule (Order of Operations): Always solve in this sequence: Brackets → Of (powers) → Division → Multiplication → Addition → Subtraction
⚡ MAT Exam Tip: Always verify your answer using reverse operations. For addition, subtract one addend; for multiplication, divide the product. This catches errors in time-pressure situations.
Mental Calculation Drills:
- $47 + 38 = 85$ | $156 - 89 = 67$ | $23 \times 7 = 161$ | $144 \div 8 = 18$
- $12^2 = 144$ | $15^2 = 225$ | $25^2 = 625$ | $35^2 = 1225$
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Basic Arithmetic Operations — Deep Dive
Integers and the Number Line
Integers extend infinitely in both directions: …, -3, -2, -1, 0, 1, 2, 3, …
Properties of Addition and Subtraction:
| Property | Addition | Subtraction |
|---|---|---|
| Commutative | $a + b = b + a$ | $a - b \neq b - a$ |
| Associative | $(a+b)+c = a+(b+c)$ | $(a-b)-c \neq a-(b-c)$ |
| Identity | $a + 0 = a$ | $a - 0 = a$ |
| Inverse | $a + (-a) = 0$ | $a - a = 0$ |
Multiplication Properties:
- Commutative: $a \times b = b \times a$
- Associative: $(a \times b) \times c = a \times (b \times c)$
- Distributive: $a \times (b + c) = a \times b + a \times c$
- Identity: $a \times 1 = a$
- Zero property: $a \times 0 = 0$
Fraction Operations
Addition/Subtraction: Find LCM of denominators, convert, then add/subtract numerators.
Example: $\frac{3}{4} + \frac{5}{6}$
- LCM(4,6) = 12
- $\frac{3}{4} = \frac{9}{12}$, $\frac{5}{6} = \frac{10}{12}$
- $\frac{9}{12} + \frac{10}{12} = \frac{19}{12} = 1\frac{7}{12}$
Multiplication: Multiply numerators together, denominators together. Simplify before multiplying if possible.
Example: $\frac{3}{4} \times \frac{8}{9} = \frac{3 \times 8}{4 \times 9} = \frac{24}{36} = \frac{2}{3}$
Division: Multiply by the reciprocal of the divisor.
Example: $\frac{3}{4} \div \frac{5}{6} = \frac{3}{4} \times \frac{6}{5} = \frac{18}{20} = \frac{9}{10}$
Percentage as a Specialized Operation
Percentage is simply fraction with denominator 100:
- $35% = \frac{35}{100} = \frac{7}{20}$
- To find $X%$ of $Y$: $\frac{X}{100} \times Y$
- To find what percent $A$ is of $B$: $\frac{A}{B} \times 100$
⚡ MAT Shortcut: To find percentage increase/decrease, use: $\frac{\text{New} - \text{Old}}{\text{Old}} \times 100$. For successive percentage changes, multiply: if something increases by 10% then 20%, overall = $1.10 \times 1.20 = 1.32$ (32% increase, NOT 30%).
Average as a Form of Division
Average = Sum of values ÷ Number of values
When all values increase by same amount $k$, average increases by $k$. When some values are removed, adjust sum accordingly.
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Basic Arithmetic Operations — Complete Theory
Modular Arithmetic (Remainder Theorems)
When dividing $a$ by $b$, we write $a = bq + r$ where $0 ≤ r < b$.
Key Modular Properties:
- $(a + b) \mod n = ((a \mod n) + (b \mod n)) \mod n$
- $(a \times b) \mod n = ((a \mod n) \times (b \mod n)) \mod n$
This is how we find unit digits of large powers (MAT favourite!):
- Find $7^{45}$ unit digit: $45 \mod 4 = 1$ → unit digit = 7
Number Base Operations
MAT sometimes tests base conversion:
- Decimal (base 10) to binary: repeatedly divide by 2
- Binary to decimal: sum $d_n \times 2^n$ for each digit
Example: $13_{10} = 1101_2$ because: $1\times2^3 + 1\times2^2 + 0\times2^1 + 1\times2^0 = 8 + 4 + 0 + 1 = 13$
Rate, Ratio, and Proportion as Derived Operations
Rate = comparison of two quantities with different units (e.g., km/hr)
Ratio = comparison of two quantities with same units (e.g., 3:5)
Proportion = equality of two ratios ($a:b = c:d$ means $a \times d = b \times c$)
⚡ MAT PYQ Pattern: Questions on chain rule (direct/indirect proportion) appear frequently. Example: “If 12 workers complete a job in 20 days, how long will 15 workers take?” → Workers and days are inversely proportional: $12 \times 20 = 15 \times x$ → $x = 16$ days.
Verification and Error-Checking Strategies
- Estimate first: Round numbers to nearest 10/100, compute approximate answer, then check if exact answer is in same ballpark.
- Digit-sum check: For addition, sum digits of addends and compare to digit-sum of result. For multiplication, digit-sum of product equals digit-sum of product of digit-sums.
- Divide to verify multiply: If $a \times b = c$, then $c \div b = a$ exactly.
- Cross-check with inverse: Use subtraction to verify addition, division to verify multiplication.
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