5.3 Four Operations & Multi-Step Calculations

Quick Summary: Fluid mastery of the four basic arithmetic operations (addition, subtraction, multiplication, and division) forms the engine of performance in the mathematics section of selective entrance tests such as the ISEB Common Pre-Test. Candidates must possess absolute fluency in formal written algorithms for multi-digit whole numbers and decimals, leverage powerful mental shortcuts to bypass time-consuming calculation, and systematically extract numerical relationships from complex, wordy problem stems.

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Column Addition and Subtraction Across Multiple Zeros

Formal column addition and subtraction provide reliable algorithms for large numbers and decimals. However, selective assessments frequently target two specific vulnerabilities:

1. Failing to align decimal points when numbers have differing decimal lengths.
2. Making errors during repeated borrowing across consecutive zeros.

Decimal Alignment: The Ragged Edge Hazard

When adding or subtracting decimals, you must align the decimal points vertically. Never right-align digits blindly as you would with whole numbers:

   INCORRECT (Right-aligned):             CORRECT (Decimal-aligned & Padded):
        2 3 . 6                                2 3 . 6 0 0
      -   7 . 8 4 2                          -   7 . 8 4 2
      ───────────────                        ─────────────
       (Calculates nonsense)                   1 5 . 7 5 8

If an empty space exists above or below a decimal digit in subtraction, you must insert a trailing zero before attempting to borrow.

The 'Cascading Borrow' vs The 'Subtract One' Strategy

Subtracting from numbers like $1,000$, $5,000$, or $10,000$ often causes arithmetic breakdowns because students must borrow across multiple columns:

Standard Decomposition:            The 'Subtract One' Trick:
    4  9  9 10                          (5,000 - 1) = 4,999
    5, 0  0  0                        - 2,367       - 2,367
  - 2, 3  6  7                        ───────       ───────
  ────────────                                        2,632  (+ 1)
    2, 6  3  3                                      = 2,633

Pro-Tip: The 'Subtract One' Mental Strategy When subtracting any number from a multiple of $1,000$ (like $5,000 - 2,367$):

1. Subtract $1$ from the top number: $5,000 - 1 = 4,999$.
2. Perform the subtraction: $4,999 - 2,367 = 2,632$. Here the hundreds, tens and units are all nines, and the thousands subtraction also works without regrouping. This shortcut is helpful in this example; it does not eliminate every possible regrouping in every subtraction.
3. Add $1$ back to the final result: $2,632 + 1 = 2,633$.

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Long Multiplication: Formal Algorithm vs Grid Method

Multiplication of two-digit and three-digit integers can be executed using either the compact formal column algorithm or the grid method. Both methods rely on the distributive law: $(a + b) \times (c + d) = ac + ad + bc + bd$.

Multiplying Multi-Digit Integers: $436 \times 27$

Method A: Formal Column Multiplication

$$\begin{array}{rl} 436 & \\ \times \quad 27 & \\ \hline 3052 & \leftarrow (436 \times 7) \\ 8720 & \leftarrow (436 \times 20 \text{; note the place-holding zero!}) \\ \hline 11772 & \leftarrow (3052 + 8720) \end{array}$$

Method B: The Grid Method (Area Model)

Partition $436$ into $(400 + 30 + 6)$ and $27$ into $(20 + 7)$:

Optional extension: multiplying two decimals

To multiply decimals like $4.36 \times 2.7$:

1. Ignore the decimal points and multiply the numbers as integers: $$436 \times 27 = 11,772$$
2. Count total decimal places across both factors:
   • $4.36$ has 2 decimal places.
   • $2.7$ has 1 decimal place.
   • Total decimal places in product = $2 + 1 = \mathbf{3}$.
3. Place the decimal point 3 places from the right in the product: $$11,772 \longrightarrow \mathbf{11.772}$$

Why does this work mathematically?

$$4.36 \times 2.7 = \left(\frac{436}{100}\right) \times \left(\frac{27}{10}\right) = \frac{436 \times 27}{1,000} = \frac{11,772}{1,000} = 11.772$$

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Short Division and Managing Remainders

Short division (the 'bus stop' method) is the most efficient algorithm for single-digit divisors. For 11+ exams, the critical skill is not merely executing the division, but interpreting the remainder appropriately for the problem context.

Executing Short Division: $1,478 \div 6$

        0  2  4  6  r 2
     ┌────────────────
   6 │  1  4 ²7 ³8

• $1 \div 6 = 0$ remainder $1$.
• $14 \div 6 = 2$ remainder $2$.
• $27 \div 6 = 4$ remainder $3$.
• $38 \div 6 = 6$ remainder $2$.

The Three Formats for Expressing Remainders

Depending on the question, the result of $1,478 \div 6$ must be presented as:

1. Integer with remainder: $246 \text{ r } 2$.
2. Mixed number: Attach the remainder over the divisor: $246 \frac{2}{6} = 246 \frac{1}{3}$.
3. Repeating decimal quotient: Annex decimal zeros to the dividend and continue dividing: $$1,478.000 \div 6 = 246.333... = 246.\dot{3}$$

Real-World Context: When to Round UP or Round DOWN

Pre-Test word problems test whether you know what physical reality dictates:

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Mental Arithmetic Shortcuts and Tactical Strategies

In a fast-paced computer-adaptive exam, candidates who write out long multiplication or division for every item run out of time. Elite performers utilise mental arithmetic shortcuts:

1. Partitioning

Break difficult multipliers into sums of simpler numbers:

$$7 \times 86 = (7 \times 80) + (7 \times 6) = 560 + 42 = 602$$ $$6 \times 3.25 = (6 \times 3) + (6 \times 0.25) = 18 + 1.5 = 19.5$$

2. Near Multiples & Compensation

When multiplying or adding numbers near powers of 10 or decade boundaries ($9, 19, 29, 99$):

• Multiplying by 19: Multiply by $20$, then subtract 1 group: $$38 \times 19 = (38 \times 20) - 38 = 760 - 38 = 722$$
• Multiplying by 99: Multiply by $100$, then subtract 1 group: $$45 \times 99 = (45 \times 100) - 45 = 4,500 - 45 = 4,455$$
• Subtracting 98: Subtract 100, then add back 2: $$423 - 98 = (423 - 100) + 2 = 323 + 2 = 325$$

3. Doubling and Halving

If one factor is even and the other ends in $5$ (or $0.5$), you can halve one factor and double the other without altering the product:

$$24 \times 15 = 12 \times 30 = 360$$ $$16 \times 35 = 8 \times 70 = 560$$ $$36 \times 2.5 = 18 \times 5 = 9 \times 10 = 90$$

4. Factorising the Multiplier

Break the multiplier into its factors and multiply sequentially:

$$75 \times 16 = 75 \times 4 \times 4 = 300 \times 4 = 1,200$$ $$45 \times 12 = 45 \times 2 \times 6 = 90 \times 6 = 540$$

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Multi-Step Word Problems, Money & Estimation

Word problems embed arithmetic calculations within realistic narratives. Success depends on a four-stage problem-solving model:

[Stage 1: Read & Filter]
Identify the target question and filter out extraneous numbers or decorative prose.

[Stage 2: Plan the Pipeline]
Determine the required chain of operations (e.g. Subtraction → Division → Multiplication).

[Stage 3: Calculate]
Execute the calculations cleanly, employing mental shortcuts where possible.

[Stage 4: Sanity Check with Estimation]
Use front-end rounding to confirm that your answer is in the correct order of magnitude.

Worked Example: Purchasing, Change & Discounts

Scenario: Maya has £30.00. She visits a school bookstore and purchases:

• $3$ spiral notebooks at £$3.45$ each
• $4$ rollerball pens at £$1.60$ each
• $1$ geometry set costing £$5.25$

The store offers a discount promotion: "Spend over £20.00 and receive £3.00 off your total bill." How much change does Maya receive when she pays £30.00 in cash?

Step-by-step Solution:

1. Calculate the cost of the notebooks: $$3 \times £3.45 = (3 \times 3) + (3 \times 0.45) = £9.00 + £1.35 = £10.35$$
2. Calculate the cost of the pens: $$4 \times £1.60 = £6.40$$
3. Sum the un-discounted basket: $$\text{Total} = £10.35 + £6.40 + £5.25 = £22.00$$
4. Check and apply the discount: Because £$22.00 > £20.00$, the £$3.00$ discount applies: $$\text{Discounted Bill} = £22.00 - £3.00 = £19.00$$
5. Calculate change: $$\text{Change} = £30.00 - £19.00 = £11.00$$
6. Estimation Check: $3 \times 3.50 = 10.50$; $4 \times 1.50 = 6.00$; geometry = $5.00$. Total $\approx 21.50 - 3.00 = 18.50$. Change $\approx 30.00 - 18.50 = 11.50$. Our exact answer of £$11.00$ is completely sound.

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Section Review Checklist

• Decimal column subtraction requires inserting zeros above blanks before borrowing.
• Use the 'Subtract One' strategy ($4,999$ instead of $5,000$) to eliminate borrowing across zeros.
• In decimal multiplication, count total decimal places across factors to locate the decimal point.
• Check whether a division word problem requires rounding UP (e.g. buses) or rounding DOWN (e.g. completed purchases).
• Use doubling-and-halving ($24 \times 15 = 12 \times 30$) to turn difficult multiplications into effortless mental math.

Addition, equality and simple rates

Column addition also needs checking. For $28,746 + 16,859$, work from the units: $6+9=15$, write 5 and carry 1; tens $4+5+1=10$, write 0 and carry 1; hundreds $7+8+1=16$, write 6 and carry 1; thousands $8+6+1=15$, write 5 and carry 1; ten-thousands $2+1+1=4$. The result is 45,605. Estimate $29,000+17,000=46,000$ and use subtraction to check the exact sum.

The equals sign means the two sides have the same value. In $13+24=12+\square$, the left side is 37, so the missing number is 25. Do not treat equals as an instruction meaning "write the answer next"; both sides can contain calculations.

For a simple rate, identify what one unit means. If 6 identical tickets cost £42, each costs £7; 9 tickets at the same rate cost £63. If a tap delivers 4 litres each minute at a constant rate, in 7 minutes it delivers 28 litres. The constant-rate condition matters: do not apply it to a situation where the rate changes.

Four-digit multiplication check

For 3,156 × 24, multiply by 20 and by 4 separately. The products are 63,120 and 12,624; adding gives 75,744. The zero in the ×20 partial product records its place value. Estimate 3,000 × 25 = 75,000 to check the scale, then verify the exact sum rather than replacing it with the estimate.