Speed Math & Calculation Bible

Calculation System: Accuracy First, Then Speed

Speed is not the opposite of working. It is the result of removing repeatable decisions. This chapter gives you a calculation system before it gives you a trick: translate words into quantities; choose a target quantity; keep units visible; estimate the answer’s size; perform a reversible calculation; and check the sign, unit and reasonableness. The methods here are original derivations from elementary arithmetic, not claimed exam rules. A rapid answer with an invisible sign error is slower than a clear answer because it has to be repaired later. Treat a shortcut as a compressed proof: you should know what it expands to, and the condition that allows the compression.

KarmSakha preparation topics covered

• Build a written calculation format that exposes sign, unit and place-value errors
• Use estimation as a check rather than an unsupported answer method
• Separate an exact method from a speed method and state the validity condition
• Use an original error ledger for calculation, reading and setup mistakes

The five-line calculation frame

Use five lines for any arithmetic word problem. Line 1: given quantities and their units. Line 2: target quantity. Line 3: relation or equation. Line 4: calculation. Line 5: reasonableness check. This frame prevents the common habit of multiplying every number in sight because it forces you to name what the answer should measure before you compute it.

For example, if a quantity rises from 80 to 100, the absolute change is 20 and the percentage increase is 20 divided by the original 80, multiplied by 100. The target line prevents the incorrect denominator 100. Your working is not “slow”; it is a visible proof that the denominator came from the original quantity.

Use a unit check at the end. A rate times a time should have a distance-like unit; a price times a number of items should have a currency-like unit. If the unit is wrong, the arithmetic may be right but the model is wrong.

• Given: numbers with units.
• Target: what must be found.
• Relation: equation before calculation.
• Check: sign, unit and approximate size.

Estimation is a guardrail, not a licence to invent

Estimate before exact calculation when numbers are awkward. Round in a direction you can track, calculate the rough size, then return to exact arithmetic. If 49.8 is multiplied by 20.2, an estimate near 50 times 20 tells you the exact product should sit near 1,000. It does not tell you the exact answer is 1,000; it tells you that an answer near 100 or 10,000 needs a recheck.

Keep the direction of rounding visible in comparisons. If both factors are rounded upward, the rough product is an upper-leaning estimate. If one factor is rounded up and another down, the direction is mixed. This is why an estimate should be described as a range or a sanity check rather than an exact proof.

The fastest safe calculation is often a complement. Instead of 49.8 times 20.2, write 50 minus 0.2 and 20 plus 0.2, expand, and verify against the estimate. The shortcut works because multiplication is distributive; if you cannot reconstruct the expansion, you should use the longer method.

• Estimate the order of magnitude before exact arithmetic.
• Track whether rounding pushed a value up or down.
• Use a complement method only when you can expand it back to the original expression.

An error ledger that changes tomorrow's method

Do not write “careless mistake” as an explanation. Name the error class. A setup error means you used the wrong base, rate or equation. A calculation error means the model was right but arithmetic failed. A reading error means you missed a word such as more, less, at least or remaining. A speed error means you had a workable method but spent too long selecting it.

Each class has a different repair. Setup errors need a target line and a formula derivation. Calculation errors need a smaller decomposition and a reciprocal check. Reading errors need you to underline comparison and time words. Speed errors need a decision rule: use exact arithmetic for small numbers, a complement for near-base values, or an estimate to eliminate impossible options before calculating.

Review the ledger after every set of ten questions. Choose the most frequent category and write one original counter-question that triggers it. The goal is to change the workflow, not to punish yourself with a score.

• Setup: wrong equation or base.
• Calculation: right equation, wrong arithmetic.
• Reading: missed qualifier.
• Speed: method selection took too long.

Solved examples

1. Given: original ₹80, new ₹100; target: percentage increase.
2. Change = 100 − 80 = 20.
3. Percentage increase = change/original × 100 = 20/80 × 100.
4. 20/80 = 1/4, so the increase is 25%.

1. Estimate: 50 × 20 = 1,000, so the exact product should be close to 1,000.
2. Write (50 − 0.2)(20 + 0.2).
3. Expand: 1000 + 10 − 4 − 0.04.
4. Exact product = 1005.96, consistent with the estimate.

Practice set

Q1. A value falls from 250 to 200. Which denominator belongs in the percentage decrease?

• 250
• 200
• 50
• 450

Q2. Which answer is a reasonable estimate for 19.9 × 5.1?

• About 100
• About 10
• About 1,000
• About 1

Q3. A rate is 60 km/h for 2.5 hours. Which target unit should the answer have?

• Kilometres
• Hours
• Kilometres per hour
• Rupees

Q4. You used the correct formula but multiplied 27 × 4 as 116. Which error category is this?

• Calculation error
• Setup error
• Reading error
• Source error

Practice answers and explanations

Q1. 250 — Percentage change is measured against the original value, 250.

Q2. About 100 — Rounding to 20 × 5 gives 100, so an exact result should be near 100.

Q3. Kilometres — Rate multiplied by time gives distance, so the result's unit should be kilometres.

Q4. Calculation error — The relation was correct; the arithmetic execution failed.

Revision checklist

• I use given, target, relation, calculation and check before I accelerate.
• I estimate for magnitude and still perform an exact calculation when exactness is required.
• I name the error type so the next revision changes a specific step.

Fractions, Ratios and Percentages: One Quantity, Three Languages

Fractions, ratios and percentages describe relationships, but they are not interchangeable until you specify the whole. A fraction a/b is a quotient. A ratio a:b is a comparison of two quantities in the same unit. A percentage is a fraction with denominator 100, normally measured against a named base. Most errors happen when a solver changes the base silently. This chapter turns the relationships into short derivations. Use a ratio when the question asks part-to-part, a fraction when it asks part-to-whole, and a percentage when it asks relative change or a share per hundred. The arithmetic is elementary; the precision lies in naming what the denominator represents.

KarmSakha preparation topics covered

• Translate between fractions, ratios, decimals and percentages
• Use a common base before comparing changes
• Derive percentage increase/decrease rather than memorising an asymmetric shortcut
• Recognise when a ratio is a part-to-part statement and when it is a part-to-whole statement

Choose the base before you calculate

Suppose red and blue balls are in the ratio 3:5. The ratio tells you that for every 3 red units there are 5 blue units. The red share of the total is not 3/5; it is 3 divided by 3 plus 5, or 3/8. The difference is a base decision: blue alone versus total balls.

A percentage question has the same discipline. If a price rises from 80 to 100, the increase is 20 over the original 80, or 25%. If it later falls from 100 to 80, the decrease is 20 over the new starting base 100, or 20%. Equal absolute changes do not create equal percentage changes because the denominators differ.

Write the denominator in words before you put it in symbols: original value, current total, selected group, or remaining quantity. This makes a ratio question resistant to wording changes.

• Ratio a:b is part-to-part.
• Share of total from a:b is a/(a+b).
• Percentage change uses the quantity from which the change is measured.

Percentage changes compose multiplicatively

An increase of p% means multiply by 1 + p/100. A decrease of p% means multiply by 1 − p/100. Two changes occur in sequence, so multiply their factors. A 20% increase followed by a 20% decrease gives 1.20 × 0.80 = 0.96, or a net 4% decrease. The common mistake is adding +20 and −20 as if both used the same base.

You can derive the net effect algebraically. Let the original be x. After p% increase it is x(1+p/100). After q% decrease it is x(1+p/100)(1−q/100). Expand only when you need the net percentage. The method works for any signs as long as you use the correct multiplication factors.

Do not use the “difference of squares” shortcut without checking that the increase and decrease percentages are equal and consecutive. Under that condition, (1+a)(1−a)=1−a² gives the net decrease. Outside that condition, use the full factors.

• Increase p% → multiply by 1 + p/100.
• Decrease p% → multiply by 1 − p/100.
• Equal consecutive increase/decrease p% → net decrease p²/100 percent.

Use ratio scaling only when the relationship is proportional

If 3 pens cost ₹45 at the same unit price, 5 pens cost ₹75 because cost is directly proportional to number of pens under the stated same-price condition. The ratio method works because the unit cost is constant. If a discount, delivery fee or bulk tier is introduced, the relation changes and simple scaling may fail.

For inverse relations, identify the constant product. If a fixed job takes 6 workers 10 days, the worker-days total is 60 under the same productivity assumption. Eight equivalent workers need 60/8 days. The shortcut workers × days = constant is not a magic rule; it follows from a fixed amount of work and equal rates.

Every proportionality shortcut needs a sentence of validity. Write “same unit price,” “fixed work, equal rates,” or “same speed.” If that sentence is false, return to a fuller equation.

• Direct proportion: one factor scales with the other under a constant unit rate.
• Inverse proportion: product stays constant only under a fixed total and equal-rate assumption.
• State the validity condition before cross-multiplying.

Solved examples

1. Take boys = 3k and girls = 5k.
2. Total = 8k.
3. Boys' share = 3k/8k.
4. Cancel k.

1. Increase factor = 1.20.
2. Decrease factor = 0.80.
3. Combined factor = 0.96.
4. Final value is 96% of original, so net change is 4% decrease.

Practice set

Q1. In a ratio 4:6, what fraction of the total is the first quantity?

• 2/5
• 4/6
• 1/6
• 4/10

Q2. A value increases 10% then increases 20%. What multiplier applies?

• 1.32
• 1.30
• 1.12
• 1.02

Q3. Six identical notebooks cost ₹90. At the same unit price, ten cost how much?

• ₹150
• ₹120
• ₹135
• ₹180

Q4. Eight equal-rate workers finish a job in 15 days. How many days for 12 equal-rate workers?

• 10
• 12
• 15
• 22.5

Practice answers and explanations

Q1. 2/5 — First share = 4/(4+6) = 4/10 = 2/5.

Q2. 1.32 — Multiply 1.10 by 1.20, giving 1.32.

Q3. ₹150 — Unit price is ₹90/6 = ₹15; ten cost ₹150.

Q4. 10 — Worker-days are fixed: 8 × 15 = 120; 120/12 = 10.

Revision checklist

• I name the base before I write a denominator.
• I multiply percentage factors for sequential changes.
• I state the constant-rate or fixed-work condition before I use a proportion shortcut.

Averages, Profit–Loss and Interest: Preserve the Total Behind the Shortcut

Averages, profit/loss and interest look like formula chapters, but they are all total-preservation chapters. An average is total divided by count. Profit percentage is profit divided by cost price, unless the question explicitly names a different measure. Simple interest grows by the same amount per period on the original principal; compound growth multiplies by a factor each period. The speed comes from keeping the base visible. Whenever a shortcut appears, ask what total it preserves and what denominator the percentage uses. If you cannot answer those two questions, use the full equation.

KarmSakha preparation topics covered

• Derive weighted average from total divided by count
• Use profit/loss percentages against the correct cost-price base
• Separate simple-interest and compound-growth models
• Check whether a shortcut preserves total, base and time period

Average is a total, not a middle-looking number

The average of n quantities is their sum divided by n. If the average of 8 values is 17, their total is 136. This reverse route is often faster than reconstructing every value. When one value changes from 12 to 19, the total changes by 7, so the average changes by 7/8—not by 7. The count controls the spread of the change.