Class 8 · English Version · Mathematics
2026 Class 8 Mathematics · English Version · Chapter 2 · Chapter companion part
Explain simple profit, use annual rates and time consistently, and recover a missing principal, rate or duration.
Class 8 English Version: part B of the chapter 2 companion. Eight teaching sections, 30 written tasks and a notebook challenge. Selected companion coverage, not complete textbook coverage; use all four parts and ask a teacher to review explanations and open solutions.
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P is the initial principal, I is accumulated simple profit, and A = P + I is the total amount, called profit-principal in the source. A deposit of 6000 gaining 1440 has amount 7440. Profit is not the whole amount. The source also uses interest terminology; this lesson follows its calculation model without recommending a financial product.
Use R for the percentage number, such as 8, and r = R/100 = 0.08 for the decimal annual rate. An 8% annual simple rate earns Tk 8 per Tk 100 of initial principal in one year. Simple profit uses the original principal each year: I = Prt and A = P(1 + rt). No profit is reinvested in this model.
When r is annual, t is in years. Three months is 3/12 = 0.25 year, six months is 0.5 year and two years six months is 2.5 years. For P = 8000, r = 0.06 and t = 0.5, I = 240. Multiplying the annual rate by six as if months were years exaggerates the time by a factor of twelve.
From I = Prt, find P = I/(rt), r = I/(Pt), or t = I/(Pr) when the denominator is nonzero. Convert r back to a percentage number using R = 100r. If amount A is known instead of I, either subtract a known P or use P = A/(1 + rt). Define the unknown and check it in the original equation.
If I is a fraction f of P, then A = (1 + f)P. For I = 3P/10 and A = 6500, P = 6500/1.3 = 5000 and I = 1500. Doubling the amount means I = P, not I = 2P. At a fixed simple rate, tripling requires twice the doubling time because the required profits are P and 2P.
Suppose a fixed principal has simple amount 5600 after three years and 6000 after five. The increase 400 over two years means 200 profit per year. Subtract three years of profit from 5600 to find P = 5000; r = 200/5000 = 0.04 = 4% annually. This constant-difference method does not apply unchanged to compound amounts.
If the same P and t use rates r1 and r2, the profit difference is P(r2 − r1)t. An increase from 8% to 10% is two percentage points, or a decimal difference 0.02. If two deposits must earn the same simple profit, equate their P × r × t values. Compare rates only after putting periods in consistent units.
Some source wording is inconsistent. In simple example 5, printed 16, the sidebar lists six years while the prompt and arithmetic use five: 1500/(3000 × 5) = 0.10, or 10%; using six would instead give 8⅓%. Exercise 2.1 item 20C asks for a principal reaching a stated amount without a duration, so it has no unique answer. Our numeric tasks specify all needed conditions.
For principal 6000 at 8% simple annual profit, the yearly addition is 480. After three years the profit is 1440 and amount 7440. Check the amount table: 6000, 6480, 6960, 7440.
Read the annual percentage as profit on each Tk 100 of initial principal, not as a fixed Tk 100 payout. Use the explicit decimal-rate convention in every formula.
These are specified classroom models. No assumption about an actual account’s rate, guarantee, fee or payout schedule follows from the calculation.
Exercise 2.1(17) gives amount Tk 5600, profit two fifths of principal and an annual simple rate of 8%. Amount is P + 2P/5 = 7P/5, so P = 5600 × 5/7 = Tk 4000. Profit is Tk 1600 and time is 1600/(4000 × 0.08) = 5 years. The printed key lists only 5 years. A complete written answer supplies both principal and time, with units.
Define the annual rate before substituting.
r = R/100 I = Prt A = P + I = P(1 + rt)
Simple profit uses the unchanged initial principal. No deposits, withdrawals or reinvestment are included in this formula.
Rearrange the equation before entering numbers, and check the result in I = Prt.
P = I/(rt) r = I/(Pt) t = I/(Pr) P = A/(1 + rt)
Division needs a nonzero denominator. Amount alone and a rate alone cannot determine principal without the time and model.
Read each duration in years for use with an annual rate. The last column uses invented principal 8000 and rate 6%.
| Duration | Years t | Simple profit (Tk) |
|---|---|---|
| 3 months | 0.25 | 120 |
| 6 months | 0.5 | 240 |
| 2 years 6 months | 2.5 | 1200 |
| 4 years | 4 | 1920 |
For six months, 8000 × 0.06 × 0.5 = 240. The percentage is annual, so six months is not six annual periods.
Use these forms only for a fixed principal with simple profit and no other transactions.
Annual profit = (A2 − A1)/(t2 − t1) P = A1 − t1 × annual profit Difference in I = P(r2 − r1)t
Do not use constant differences to infer an annual compound rate; the compound base changes.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original invented values; all bars start at zero and share one scale of Tk 25 per drawing unit. No deposits or withdrawals are included.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original invented values; the two segment lengths are proportional to 5000 and 1500. Labels define the quantities independently of colour.
Show your working on paper. Check your answer, then choose Next.
0.08
8/100 = 0.08.
0.25
3/12 = 0.25.
2.5
2 + 6/12 = 2.5.
1440
6000 × 0.08 × 3 = 1440.
7440
6000 + 1440 = 7440.
6000
1200/(0.05 × 4) = 6000.
6
100 × 900/(5000 × 3) = 6.
3
1800/(6000 × 0.10) = 3.
6000
7800/(1 + 0.10 × 3) = 6000.
5000
P + 0.3P = 6500; P = 5000.
12.5
I = P, so r = 1/8 = 0.125, or 12.5%.
12
Twice as much profit requires twice the time.
5000
Annual profit is 200; 5600 − 3 × 200 = 5000.
1000
80/(0.02 × 4) = 1000.
6000
The amount factor is 5200/4000 = 1.3; 7800/1.3 = 6000.
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Profit is the extra 1440; amount includes the original 6000, so 6000 + 1440 = 7440.
Name all three quantities and use consistent units.
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R percent means R per 100, or r = R/100. One year earns Pr; t years earn Prt on the unchanged original principal.
Mention the simple-model condition and annual time unit.
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t = 2.5; I = 7200 × 0.075 × 2.5 = 1350; A = 8550.
Check conversion, decimal rate and amount versus profit.
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P = 9600/(1 + 0.05 × 4) = 8000; I = 1600. Check 8000 × 0.05 × 4 = 1600.
Do not insert amount as principal into I = Prt.
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P = 2040/(0.085 × 4) = 6000.
Read the fractional percentage as 0.085, not 8.5.
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r = 1440/(4800 × 5) = 0.06; R = 6%.
State the rate as both decimal and percentage.
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t = 1500/(5000 × 0.12) = 2.5 years = two years six months.
Require a reverse check and a time unit.
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1.4P = 8400 gives P = 6000 and I = 2400. t = 2400/(6000 × 0.08) = 5 years.
Keep the fraction relative to principal, not amount.
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Doubling requires profit P in five years; tripling requires 2P, reached in ten years at the same simple rate. Fifteen years would earn 3P and give amount 4P.
Separate profit multiples from amount multiples.
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Annual profit = (7500 − 6900)/2 = 300. Principal = 6900 − 900 = 6000. Rate = 300/6000 = 5% per year.
Explain the two-year gap and check both amounts.
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Difference = 6000 × (0.09 − 0.07) × 3 = 360.
Use a two-percentage-point difference, not 2% of the old profit.
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Target profit is 1000 × 0.06 × 5 = 300. Required P = 300/(0.05 × 2.5) = 2400.
Equate profits after converting time.
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Use rate 0.06 and time 0.5 year. I = 8000 × 0.06 × 0.5 = 240.
Identify separate rate and time errors.
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The question and calculation use five years, giving 10% annually. Six years would give 8⅓%. State the inconsistency instead of silently combining incompatible inputs.
Accept either conditional result only with its duration explicitly named.
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P = 9000/(1 + 0.05t). At t = 2, P = 9000/1.1; at t = 4, P = 7500. Different times give different principals, so a duration is needed.
A formula with the missing variable is better than a guessed single answer.
Make a formula page and solve one task for each missing value P, r, t and I.
Keep your work on paper or in a local file. This page does not upload it or automatically grade the project.
This original Kidemy practice follows the topic order of 2026 Class 8 Mathematics · English Version, chapter 2, printed pages 12–28. It does not replace the complete chapter or its practical teaching.
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