Chapter Review

Arithmetic Business Applications

Percentage Applications · Profit, Loss, and Discount · Simple and Compound Interest · Time, Work, and Distance Problems

Percentage Fundamentals

A percentage expresses a ratio as a fraction of 100, enabling direct comparison of different quantities. The base (denominator) must be identified correctly in every problem — it is usually the original or total value.

Key Points

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Percentage = (Part / Whole) × 100
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Any x% is equivalent to decimal x/100 — use as a multiplier
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Reverse percentage: if value V is after p% change, original = V / (1 ± p/100)
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Equal increase and decrease of p% never returns to original — net loss is (p²/10000) of original
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Percentage of a percentage: x% of y% = (x × y / 100)%

Successive and Compound Percentage Changes

When multiple percentage changes occur in sequence, each applies to the result of the previous one. The net effect is found by multiplying individual multipliers, not by adding percentages.

Key Points

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Net % change for two successive changes: a + b + (ab/100)
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Multiplier method: final = V × (1 + a/100)(1 + b/100) — works for any number of changes
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Population/compound growth: P_final = P_initial × (1 + r/100)^n
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The cross term ab/100 is what makes successive changes differ from simple addition
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Order of successive changes does not affect the final result

Formula

$$\text{Net \% Change} = a + b + \frac{ab}{100}$$

Profit, Loss, and the Price Triangle

Every transaction involves cost price (CP), marked price (MP), and selling price (SP). Profit occurs when SP > CP, loss when CP > SP. All percentage calculations use CP as the base, except discount which uses MP.

Key Points

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Profit = SP − CP; Loss = CP − SP — always measured against CP
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Profit % = (SP − CP)/CP × 100; Loss % = (CP − SP)/CP × 100
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Markup % = (MP − CP)/CP × 100 — sets the tag price above cost
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Discount % = (MP − SP)/MP × 100 — calculated on MP, not CP
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Chain: SP = CP × (1 + Markup%/100) × (1 − Discount%/100)
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Equal % profit and % loss on same SP always gives net loss of (a/10)²%

Formula

$$\text{SP} = \text{MP}\left(1 - \frac{d}{100}\right) = \text{CP}\left(1 + \frac{p}{100}\right)$$

Successive Discounts

Multiple discounts applied in sequence each operate on a shrinking base, so the effective discount is always less than the arithmetic sum of individual percentages.

Key Points

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Two-discount shortcut: effective = a + b − (ab/100)
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Three-discount formula: a + b + c − (ab+bc+ca)/100 + abc/10000
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SP = MP × (1 − a/100)(1 − b/100)(1 − c/100)
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Order of successive discounts does not matter (commutative)
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To find CP from SP given profit/loss %: CP = (SP × 100)/(100 ± p)

Formula

$$\text{Effective discount} = a + b - \frac{ab}{100}$$

Simple Interest

Simple interest is computed only on the original principal, producing linear growth. The interest amount is the same every period, making it directly proportional to P, R, and T individually.

Key Points

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SI = (P × R × T) / 100 — R in percentage form, T in years
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Amount A = P + SI = P(1 + RT/100)
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Convert months to years (÷12), days to years (÷365) before using the formula
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Doubling any one of P, R, or T doubles the interest
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Finding P from amount: P = (A × 100)/(100 + RT)

Formula

$$SI = \frac{P \times R \times T}{100}$$

Compound Interest and Effective Rate

Compound interest is calculated on both the principal and accumulated interest, producing exponential growth. More frequent compounding increases returns, captured by the effective annual rate.

Key Points

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Annual: A = P(1 + R/100)^T; General: A = P(1 + R/(n×100))^(nT)
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CI = A − P — always compute A first, then subtract
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CI ≥ SI for same P, R, T (equal only when T ≤ 1)
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2-year difference shortcut: CI − SI = P(R/100)²
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Effective rate: R_eff = (1 + R/(n×100))^n − 1 — always ≥ nominal rate
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For fractional years: compound full years, then SI on the remainder

Formula

$$A = P\left(1 + \frac{R}{n \times 100}\right)^{nT}$$

Work Rate and Combined Work

Work problems reduce to W = R × T. When treating a job as one unit, each worker's rate is 1/T. Combined rates add, making the LCM method the fastest solving approach.

Key Points

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Set W = 1 for one complete job; rate R = 1/T where T is solo completion time
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Combined rate: R_total = R₁ + R₂ + R₃ + …
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Two-worker shortcut: T_together = (t₁ × t₂)/(t₁ + t₂)
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LCM method: set total work = LCM of individual times, express rates as integer units/day
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Efficiency ratio equals the inverse of the time ratio
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Man-days: total work = workers × days — useful for workforce scaling

Formula

$$T_{\text{together}} = \frac{t_1 \cdot t_2}{t_1 + t_2}$$

Speed, Distance, Time and Relative Speed

The speed-distance-time triangle (s = d/t) underpins all motion problems. Relative speed determines how fast two objects approach or separate, depending on their direction of travel.

Key Points

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Three forms: s = d/t, d = s×t, t = d/s — rearrange as needed
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Average speed = total distance / total time (NOT average of speeds)
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Equal distances: avg speed = 2ab/(a+b) (harmonic mean)
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Relative speed: add for opposite directions, subtract for same direction
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Train crossing: time = (L₁ + L₂) / s_rel — convert km/h to m/s via ×5/18
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Platform crossing: distance = train length + platform length

Formula

$$s_{\text{rel}} = s_1 + s_2 \text{ (opposite)}, \quad s_{\text{rel}} = |s_1 - s_2| \text{ (same)}$$

Boats, Streams, and Circular Tracks

Boat speed is modified by the current: boosted downstream, reduced upstream. On circular tracks, two runners meet when the faster gains one full lap, governed by relative speed.

Key Points

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Downstream speed = b + c; Upstream speed = b − c
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Boat speed b = (s_down + s_up)/2; Stream speed c = (s_down − s_up)/2
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Stream speed must be less than boat speed or upstream travel is impossible
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Circular track (same direction): t_meet = L / |s₁ − s₂|
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Circular track (opposite): t_meet = L / (s₁ + s₂)
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Pipes and cisterns: net rate = sum of fill rates − sum of drain rates

Formula

$$b = \frac{s_d + s_u}{2}, \quad c = \frac{s_d - s_u}{2}$$

Formulas

Successive Percentage Change

Net effect of two percentage changes applied in sequence

Formula

$$a + b + \frac{ab}{100}$$

Profit / Loss Percentage

Profit or loss as a fraction of cost price

Formula

$$\text{Profit \%} = \frac{\text{SP} - \text{CP}}{\text{CP}} \times 100$$

Simple Interest

Interest on the original principal only — linear growth

Formula

$$SI = \frac{P \times R \times T}{100}$$

Compound Interest (General)

Interest on principal plus accumulated interest — exponential growth with any compounding frequency

Formula

$$A = P\left(1 + \frac{R}{n \times 100}\right)^{nT}$$

CI − SI (2 Years)

Difference between compound and simple interest for a 2-year period

Formula

$$CI - SI = P\left(\frac{R}{100}\right)^2$$

Average Speed (Equal Distances)

Harmonic mean of two speeds over equal distances

Formula

$$\text{Avg Speed} = \frac{2ab}{a+b}$$

Combined Work Time

Time for two workers to complete a job together

Formula

$$T_{\text{together}} = \frac{t_1 \cdot t_2}{t_1 + t_2}$$

Boat and Stream Speeds

Extract boat speed and stream speed from downstream/upstream measurements

Formula

$$b = \frac{s_d + s_u}{2}, \quad c = \frac{s_d - s_u}{2}$$