Number, Fractions, and Approximation: JAMB Mathematics Notes

A step-by-step guide to number properties, fractions, ratio, percentages, number bases, approximation, indices, logarithms, and surds.

Number Properties & Place Value

Start by reading the number carefully. A digit does not have the same value everywhere. Its place tells you its value.

In $5.072$ , the 5 is in the units place, 0 is in the tenths place, 7 is in the hundredths place, and 2 is in the thousandths place. Therefore the value of the 7 is $0.07$ , not 7.

A prime number has exactly two positive factors: 1 and itself. The first primes are $2, 3, 5, 7, 11, 13, \dots$ . Remember that 1 is not prime.

To find an H.C.F., ask: “What is the greatest factor shared by these numbers?” To find an L.C.M., ask: “What is the first positive multiple the numbers share?” Do not mix them up.

Example 1: What is the place value of 6 in $24.063$ ?

Step 1: Move to the right of the decimal point: tenths, hundredths, thousandths.

Step 2: The 6 is in the hundredths position.

Therefore, its value is $0.06$ , or six hundredths.

Example 2: Write $84$ as a product of prime factors.

Divide by the smallest prime that works: $84 = 2 \times 42$ .

Continue: $42 = 2 \times 21$ , and $21 = 3 \times 7$ .

So $84 = 2^{2} \times 3 \times 7$ .

Example 3: Find the H.C.F. of $36$ and $48$ .

Prime factors: $36 = 2^{2} \times 3^{2}$ and $48 = 2^{4} \times 3$ .

Take only the prime factors common to both, using the smaller power: $2^{2} \times 3 = 12$ .

So the H.C.F. is 12.

Example 4: Find the L.C.M. of $12$ and $18$ .

Prime factors: $12 = 2^{2} \times 3$ and $18 = 2 \times 3^{2}$ .

For the L.C.M., take every required prime using the larger power: $2^{2} \times 3^{2} = 36$ .

So the L.C.M. is 36.

Fractions & Mixed Numbers

A fraction $\frac{a}{b}$ has a numerator on top and a denominator below. The denominator cannot be zero.

When adding or subtracting fractions, first make the denominators the same. When multiplying, multiply top by top and bottom by bottom. When dividing by a fraction, multiply by its reciprocal: turn the second fraction upside down.

A mixed number such as $2 \frac{1}{3}$ means two whole units and one third. Convert it to an improper fraction before a long calculation: $2 \frac{1}{3} = \frac{2 ( 3 ) + 1}{3} = \frac{7}{3}$ .

The word of usually means multiply. For example, $\frac{3}{5}$ of 40 means $\frac{3}{5} \times 40$ .

Example 5: Simplify $\frac{18}{24}$ .

The H.C.F. of 18 and 24 is 6.

Divide top and bottom by 6: $\frac{18}{24} = \frac{3}{4}$ .

Example 6: Find $\frac{2}{3} + \frac{5}{8}$ .

The L.C.M. of 3 and 8 is 24.

Write both fractions with denominator 24: $\frac{2}{3} = \frac{16}{24}$ and $\frac{5}{8} = \frac{15}{24}$ .

Add: $\frac{16}{24} + \frac{15}{24} = \frac{31}{24} = 1 \frac{7}{24}$ .

Example 7: Calculate $2 \frac{1}{4} \times 1 \frac{2}{3}$ .

Convert first: $2 \frac{1}{4} = \frac{9}{4}$ and $1 \frac{2}{3} = \frac{5}{3}$ .

Multiply: $\frac{9}{4} \times \frac{5}{3} = \frac{45}{12} = \frac{15}{4} = 3 \frac{3}{4}$ .

Example 8: Calculate $\frac{3}{5} \div \frac{9}{10}$ .

Keep the first fraction and turn the second upside down: $\frac{3}{5} \times \frac{10}{9}$ .

Cancel common factors or multiply: $\frac{30}{45} = \frac{2}{3}$ .

Ratio, Proportion & Rates

A ratio compares quantities in a stated order. In $2 : 3$ , the first quantity has 2 parts for every 3 parts of the second. If the total is shared in a ratio, add the ratio parts first.

A proportion says that two ratios are equal. A rate compares quantities with different units, such as kilometres per hour. Always check the units before calculating.

For speed questions, use $speed = \frac{distance}{time}$ . From the same triangle, $distance = speed \times time$ and $time = \frac{distance}{speed}$ .

Example 9: Share ₦45,000 in the ratio $2 : 3$ .

Total ratio parts: $2 + 3 = 5$ .

One part is $45000 \div 5 = 9000$ .

The shares are $2 (9000) = ₦18, 000$ and $3 (9000) = ₦27, 000$ .

Example 10: If 5 books cost ₦6,000 at the same rate, what do 8 books cost?

First find one book: $6000 \div 5 = 1200$ .

Then 8 books cost $8 \times 1200 = ₦9, 600$ .

Example 11: A car travels $180 km$ in 3 hours. Find its average speed.

Use $speed = \frac{distance}{time}$ .

So $speed = \frac{180}{3} = 60 km/h$ .

Example 12: A journey takes 4 hours at $60 km/h$ . How long would the same distance take at $80 km/h$ ?

First find the distance: $60 \times 4 = 240 km$ .

Then new time is $\frac{240}{80} = 3$ hours.

Percentages & Percentage Error

Percent means “out of 100”. Thus $35% = \frac{35}{100} = 0.35$ .

To find a percentage of a quantity, multiply by the percentage written as a fraction or decimal. For percentage increase or decrease, the denominator is the original value.

$percentage change = \frac{change}{original} \times 100%$ .

For measurement error, first take the positive difference between measured and actual values. Then compare that error with the actual value:

$percentage error = \frac{∣ measured - actual ∣}{actual} \times 100%$ .

Example 13: Find $15%$ of ₦24,000.

Write 15% as $\frac{15}{100}$ .

Then $\frac{15}{100} \times 24000 = 3600$ .

So the answer is ₦3,600.

Example 14: A price rises from ₦80 to ₦100. Find the percentage increase.

Increase: $100 - 80 = 20$ .

Use the original value 80: $\frac{20}{80} \times 100% = 25%$ .

Example 15: After a 20% discount, a shirt costs ₦12,000. What was the original price?

After a 20% discount, 80% of the original remains.

Let the original price be $P$ . Then $0.8 P = 12000$ .

So $P = 12000 \div 0.8 = ₦15, 000$ .

Example 16: An actual length is $50 cm$ , but a measurement gives $51 cm$ . Find the percentage error.

Absolute error: $∣51 - 50∣ = 1 cm$ .

Percentage error: $\frac{1}{50} \times 100% = 2%$ .

Number Bases

The small number written at the lower right tells you the base. For example, $101 1_{2}$ is read “one zero one one, base two”.

Digits must be smaller than the base. Base 2 uses only 0 and 1. Base 5 uses 0, 1, 2, 3 and 4. Therefore a digit 7 cannot appear in a base-5 number.

To convert to base 10, multiply each digit by its place value. Place values are powers of the base, starting from power 0 at the right.

Example 17: Convert $101 1_{2}$ to base 10.

Read the place values: $2^{3}, 2^{2}, 2^{1}, 2^{0}$ .

So $101 1_{2} = 1 (8) + 0 (4) + 1 (2) + 1 (1) = 11$ .

Therefore, $101 1_{2} = 1 1_{10}$ .

Example 18: Convert $2 5_{10}$ to base 2.

Repeatedly divide by 2 and record remainders: $25 \div 2 = 12$ remainder 1; $12 \div 2 = 6$ remainder 0; $6 \div 2 = 3$ remainder 0; $3 \div 2 = 1$ remainder 1; $1 \div 2 = 0$ remainder 1.

Read the remainders from bottom to top: $1100 1_{2}$ .

Example 19: Convert $24 3_{5}$ to base 10.

Use powers of 5: $24 3_{5} = 2 (5^{2}) + 4 (5^{1}) + 3 (5^{0})$ .

So $24 3_{5} = 50 + 20 + 3 = 7 3_{10}$ .

Example 20: Add $10 1_{2} + 1 1_{2}$ .

Convert to decimal if that feels safer: $10 1_{2} = 5$ and $1 1_{2} = 3$ .

The sum is 8, and $8_{10} = 100 0_{2}$ .

So $10 1_{2} + 1 1_{2} = 100 0_{2}$ .

Standard Form & Approximation

Standard form is $a \times 1 0^{n}$ , where $1 \leq a < 10$ . There must be exactly one non-zero digit before the decimal point in $a$ .

Moving the decimal point left to make $a$ gives a positive power of 10 for a large number. Moving it right for a small number gives a negative power.

For significant figures, start counting from the first non-zero digit. For decimal places, count positions after the decimal point. These are not the same instruction.

Example 21: Write $5, 720, 000$ in standard form.

Move the decimal point so the first part is 5.72. It moved 6 places left.

Therefore, $5, 720, 000 = 5.72 \times 1 0^{6}$ .

Example 22: Write $0.000438$ in standard form.

Move the decimal point right until the number is 4.38. It moved 4 places.

Because the original number is small, the power is negative.

So $0.000438 = 4.38 \times 1 0^{- 4}$ .

Example 23: Round $0.007864$ to 3 significant figures.

Ignore the leading zeros when counting significant figures. The first three significant digits are 7, 8 and 6.

The next digit is 4, so 6 stays unchanged.

The answer is $0.00786$ .

Example 24: Evaluate $(3.0 \times 1 0^{5}) (2.0 \times 1 0^{- 3})$ in standard form.

Multiply the ordinary numbers: $3.0 \times 2.0 = 6.0$ .

Add the powers: $1 0^{5} \times 1 0^{- 3} = 1 0^{2}$ .

So the answer is $6.0 \times 1 0^{2}$ .

Indices & Exponential Equations

In $a^{n}$ , $a$ is the base and $n$ is the index or exponent. Read $2^{4}$ as “2 to the power 4”, meaning $2 \times 2 \times 2 \times 2$ .

For the same non-zero base:

$a^{m} \times a^{n} = a^{m + n}$
$a^{m} \div a^{n} = a^{m - n}$
$(a^{m})^{n} = a^{mn}$
$a^{0} = 1$
$a^{- n} = \frac{1}{a ^{n}}$
$a^{1/ n} = n a$

For an exponential equation, try to write both sides with the same base. Then compare the exponents.

Example 25: Simplify $2^{3} \times 2^{5}$ .

The base is the same, so add the exponents: $2^{3 + 5} = 2^{8} = 256$ .

Example 26: Simplify $12 5^{- 1/3}$ .

A negative power means reciprocal: $12 5^{- 1/3} = \frac{1}{12 5 ^{1/3}}$ .

The power $1/3$ means cube root, and $3125 = 5$ .

So the answer is $\frac{1}{5}$ .

Example 27: Solve $3^{x + 1} = 81$ .

Write 81 with base 3: $81 = 3^{4}$ .

Now $3^{x + 1} = 3^{4}$ , so $x + 1 = 4$ .

Therefore, $x = 3$ .

Example 28: Simplify $\frac{5 ^{7}}{5 ^{3}}$ .

The base is the same, so subtract exponents: $5^{7 - 3} = 5^{4} = 625$ .

Logarithms

A logarithm is a power written another way. This is the most important sentence in this section:

$lo g_{b} a = c$ means exactly the same thing as $b^{c} = a$ .

For example, $lo g_{2} 8 = 3$ because $2^{3} = 8$ . The small lower number 2 is the base of the logarithm.

Useful laws, for positive values where the logarithms are defined, are:

$lo g_{b} (M N) = lo g_{b} M + lo g_{b} N$
$lo g_{b} (M / N) = lo g_{b} M - lo g_{b} N$
$lo g_{b} (M^{k}) = k lo g_{b} M$
$lo g_{b} 1 = 0$ and $lo g_{b} b = 1$

Example 29: Evaluate $lo g_{3} 81$ .

Ask: “3 to what power gives 81?”

Since $3^{4} = 81$ , $lo g_{3} 81 = 4$ .

Example 30: Solve $lo g_{2} x = 5$ .

Change from logarithmic form to exponential form: $2^{5} = x$ .

Therefore, $x = 32$ .

Example 31: Simplify $lo g_{10} 1000 - lo g_{10} 10$ .

$lo g_{10} 1000 = 3$ because $1 0^{3} = 1000$ .

$lo g_{10} 10 = 1$ .

Therefore, the answer is $3 - 1 = 2$ .

Example 32: If $lo g_{10} 2 = 0.3010$ , find $lo g_{10} 20$ .

Write $20 = 2 \times 10$ .

Use the product law: $lo g_{10} 20 = lo g_{10} 2 + lo g_{10} 10$ .

So $lo g_{10} 20 = 0.3010 + 1 = 1.3010$ .

Surds & Rationalisation

A surd is an exact root such as $2$ or $35$ . Do not turn it into a decimal unless the question asks for an approximation.

To simplify a square root, look for a perfect-square factor. For example, $72 = 36 \times 2 = 62$ .

You may add or subtract like surds only: $32 + 52 = 82$ , but $2 + 3$ cannot be combined.

To rationalise a denominator containing one surd, multiply top and bottom by that surd. For a denominator such as $a + b$ , multiply by its conjugate $a - b$ .

Example 33: Simplify $50$ .

Find the largest square factor: $50 = 25 \times 2$ .

So $50 = 252 = 52$ .

Example 34: Simplify $38 + 218$ .

First simplify each surd: $8 = 22$ and $18 = 32$ .

So $3 (22) + 2 (32) = 62 + 62 = 122$ .

Example 35: Rationalise $\frac{5}{3}$ .

Multiply top and bottom by $3$ : $\frac{5 3}{3 3}$ .

Since $33 = 3$ , the answer is $\frac{5 3}{3}$ .

Example 36: Rationalise $\frac{1}{2 + 3}$ .

The conjugate of $2 + 3$ is $2 - 3$ .

Multiply top and bottom by it: $\frac{2 - 3}{( 2 + 3 ) ( 2 - 3 )}$ .

The denominator is $4 - 3 = 1$ .

Therefore, the answer is $2 - 3$ .

A simple exam plan

Read what the notation means before choosing a formula.
For fractions, make denominators equal only when adding or subtracting.
For ratio, keep the quantities in the order given.
For percentage change, divide by the original value.
For percentage error, divide the absolute error by the actual value.
For number bases, check that every digit is smaller than the base.
For significant figures, start counting at the first non-zero digit.
For indices, check whether the bases are the same before combining powers.
For logarithms, translate $lo g_{b} a = c$ into $b^{c} = a$ when you feel stuck.
For surds, simplify square factors before adding or rationalising.

Work one line at a time and check whether the size of your answer is reasonable.

Number, Fractions, and Approximation