Rational and Non-Rational Numbers and Compound Interest – JSS3 Mathematics Lesson Note (Third Term)

A rational number can be written as an exact fraction of two integers, while a non-rational number cannot. Compound interest is interest worked out on the principal plus the interest already earned.

Behavioural objectives

1. Define rational and non-rational numbers and give examples of each.
2. Find square roots of numbers that are not perfect squares using tables.
3. Calculate simple interest and compound interest, and the amount, on a given sum.
4. Solve problems on direct, inverse, joint and partial variation.

What are rational numbers?

A rational number is a number that can be written as the fraction p/q, where p and q are integers and q is not zero.

• Whole numbers and integers: 8 = 8/1 and 15 = 15/1.
• Fractions and mixed numbers: 3 1/2 = 7/2.
• Terminating decimals: 0.31 = 31/100 and 0.1 = 1/10.
• Recurring decimals: 17/99 = 0.171717…, 4/11 = 0.363636… and 10/3 = 3.3333…
• Square roots of perfect squares: the square root of 16/9 is 4/3.

• A recurring decimal is written with dots over the digits that repeat, for example 0.17 with dots over 1 and 7.

What are non-rational numbers?

A non-rational (irrational) number cannot be written as an exact fraction or a recurring decimal.

• Decimal form: the decimal goes on forever and never repeats.
• Square roots of non-perfect squares: √7 = 2.645751…, √17 = 4.1231056… and √29 = 5.3851648…
• Pi: π is also non-rational.

What is the difference between rational and non-rational numbers?

How do you find the square root of a number that is not a perfect square?

Use the table of square roots in the four-figure tables, or use trial and error.

1. If the number is between 1 and 100, read the square root straight from the table.
2. If the number is larger, write it as a number between 1 and 10 multiplied by 100 (or 10 000).
3. Read the square root of the small number from the table.
4. Multiply by 10 (for 100) or by 100 (for 10 000).
5. Round to the places or figures asked for.

Example 1

Find √17 to 3 significant figures.

• The table gives √17 = 4.123.
• Answer: 4.12 to 3 s.f.

Example 2

Find √293 to the nearest tenth.

1. 293 = 2.93 × 100.
2. √293 = √2.93 × √100 = 1.712 × 10 = 17.12.
3. Answer: 17.1 to the nearest tenth.

What is compound interest?

Compound interest is interest calculated each year on the principal plus all the interest already added.

• Principal (P): the money borrowed or invested at the start.
• Rate (R): the percentage interest each year.
• Time (n): the number of years.
• Amount (A): the principal plus the interest.

What is simple interest?

Simple interest is interest calculated on the original principal only, every year.

• Formula: I = PRT/100, where T is the time in years.
• Example: N2,000 at 5% for 3 years gives (2000 × 5 × 3)/100 = N300.
• Amount: A = P + I.

What is the formula for compound interest?

• Amount: A = P(1 + R/100)ⁿ.
• Compound interest: C.I. = A – P.

How do you work out compound interest?

1. Write down P, R and n.
2. Work out 1 + R/100 as a decimal.
3. Raise it to the power n.
4. Multiply by P to get the amount A.
5. Subtract P to get the compound interest.

Example 3: year by year

Find the compound interest on N10,000 for 2 years at 5%.

• Compound interest = 11,025 – 10,000 = N1,025.

Example 4: using the formula

Find the amount and compound interest on N5,000 for 3 years at 10%.

1. A = 5000 × (1.1)³ = 5000 × 1.331 = N6,655.
2. C.I. = 6,655 – 5,000 = N1,655.

What is the difference between simple and compound interest?

What is direct variation?

Direct variation describes two quantities that increase or decrease together in the same proportion.

• Statement: P varies directly as R is written P ∝ R.
• Equation: P = kR, where k is the constant of variation.
• Constant: k = P/R.
• Graph: a straight line through the origin.

Example 5

p varies directly as the square of q, and p = 27 when q = 3. Find the law, p when q = 2, and q when p = 48.

1. p = kq².
2. k = 27/3² = 27/9 = 3, so the law is p = 3q².
3. When q = 2: p = 3 × 4 = 12.
4. When p = 48: q² = 48/3 = 16, so q = 4.

Example 6: graph of direct variation

• Distance divided by time is always 5, so D = 5T.
• Plot time across and distance up.
• The points lie on a straight line through the origin.

What is inverse variation?

Inverse variation describes two quantities where one increases as the other decreases, so their product stays constant.

• Statement: T varies inversely as S is written T ∝ 1/S.
• Equation: T = k/S.
• Constant: k = TS.
• Graph: a curve, not a straight line.

Example 7

T is inversely proportional to S, and T = 2 when S = 60. Find the relationship and T when S = 90.

1. k = TS = 2 × 60 = 120.
2. The relationship is T = 120/S.
3. When S = 90: T = 120/90 = 4/3 = 1 1/3.

Example 8: graph of inverse variation

• Speed × time = 40 each time, so S = 40/t.
• Plotting the points gives a smooth curve that falls as time increases.

What is joint variation?

Joint variation is when a quantity varies with more than one other quantity, directly, inversely or both.

• Both direct: P ∝ QG means P = kQG.
• Direct and inverse: H ∝ Y/M means H = kY/M.

Example 9

H ∝ Y/M, and H = 42 when Y = 7 and M = 3. Find the relation, and H when Y = 5 and M = 9.

1. k = HM/Y = (42 × 3)/7 = 18.
2. The relation is H = 18Y/M.
3. When Y = 5 and M = 9: H = (18 × 5)/9 = 10.

Example 10: universal gas law

V varies directly as T and inversely as P. At T = 425 K and P = 1000 N/m², V = 0.255 m³. Find the formula, and P when T = 720 K and V = 0.018 m³.

1. V = kT/P, so k = PV/T = (1000 × 0.255)/425 = 255/425 = 3/5.
2. The formula is V = 3T/(5P).
3. Make P the subject: P = 3T/(5V) = (3 × 720)/(5 × 0.018) = 2160/0.09.
4. P = 24,000 N/m².

What is partial variation?

Partial variation is when a quantity is made of two parts: one part is constant and the other part varies.

Everyday examples

• Hairdresser’s charge: M = C + bT, where C is the cost of the wool, T is the time taken, and C and b are constants.
• Prepaid electricity bill: B = 750 + bU, where N750 is the fixed rental charge and U is the units used.

Common forms

• Partly constant, partly varies as G: W = a + bG.
• Partly as P, partly inversely as Q: V = aP + b/Q.

• The constants a and b are found by solving two equations at the same time.

Example 11

x is partly constant and partly varies as y². x = 3 when y = 4, and x = 1 when y = 5. Find the law and x when y = 2.

1. x = a + by².
2. Equation (i): a + 16b = 3. Equation (ii): a + 25b = 1.
3. Subtract (ii) from (i): -9b = 2, so b = -2/9.
4. From (i): a = 3 + 32/9 = 59/9 = 6 5/9.
5. The law is x = 59/9 – (2/9)y².
6. When y = 2: x = 59/9 – 8/9 = 51/9 = 5 2/3.

Example 12

T varies partly as V and partly as V³. T = 30 when V = 2, and T = 15 when V = 3. Find the law and T when V = 4.

1. T = aV + bV³.
2. 30 = 2a + 8b, which gives a + 4b = 15 (i).
3. 15 = 3a + 27b, which gives a + 9b = 5 (ii).
4. Subtract (i) from (ii): 5b = -10, so b = -2.
5. From (i): a = 15 – 4(-2) = 23.
6. The law is T = 23V – 2V³.
7. When V = 4: T = 92 – 128 = -36.

Example 13

The cost of making a dress is partly constant and partly varies with the hours taken. It costs N2,700 for 3 hours and N3,100 for 5 hours. Find the cost for 1 1/2 hours.

1. C = a + bT.
2. 2700 = a + 3b and 3100 = a + 5b.
3. Subtract: 2b = 400, so b = 200.
4. a = 2700 – 600 = 2100.
5. The law is C = 2100 + 200T.
6. When T = 1.5: C = 2100 + 300 = N2,400.

Evaluation

1. Which of these is non-rational: 3/7, 0.25, √13 or 0.6666…?
2. Find the square root of 7562 to one decimal place, using tables.
3. Find the compound interest on N8,000 borrowed for 2 years at 20%.
4. Find the simple interest and the amount on N7,000 for 4 years at 7 1/2%.
5. The current I in a circuit varies inversely as the resistance R. A current of 10 A flows when R = 20 ohms. Find I when R = 80 ohms.

Assignment

Find the amount and the compound interest on N20,000 invested for 2 years at 10%, then find √5720 to one decimal place.

• Reading: Essential Mathematics for J.S.S. 3 by Oluwasanmi A.J.S. (2014 edition), pages 38-45 and 49-53.
• Practice: Essential Mathematics Workbook for J.S.S. 3, Exercise 7.1 (numbers 1-5), Exercise 7.2 (numbers 1-12) and Exercise 5.1 (numbers 6-10).