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Reading the columns from the right gives units, tens, hundreds, thousands, ten thousands, so the digit 6 in 3 264 051 sits in the ten-thousands column. Its value is therefore 6 × 10 000 = 60 000.
Multiplication is carried out before addition, and the product of two negative numbers is positive, so (−4) × (−3) = 12. Adding the remaining term gives 12 + (−5) = 7.
A prime number has exactly two factors, 1 and itself. The number 59 is not divisible by 2, 3, 5 or 7, whereas 51 = 3 × 17, 57 = 3 × 19 and 63 = 7 × 9 all have further factors.
Rewrite every fraction over the lowest common denominator 12: 2/3 = 8/12, 3/4 = 9/12 and 1/6 = 2/12. Then 8/12 + 9/12 − 2/12 = 15/12, which simplifies to 5/4.
The profit is P300 − P250 = P50, and percentage profit is always calculated on the cost price. Hence the profit is 50/250 × 100% = 20%.
The ratio contains 3 + 4 + 5 = 12 equal parts, so each part is worth P840 ÷ 12 = P70. The largest share is 5 parts, that is 5 × P70 = P350.
Standard form is written as A × 10ⁿ where A lies between 1 and 10. Moving the decimal point five places to the right in 0.000 046 gives 4.6, so the number is 4.6 × 10⁻⁵.
Write each number in prime factors: 12 = 2² × 3, 18 = 2 × 3² and 24 = 2³ × 3. The LCM uses the highest power of each prime, so it is 2³ × 3² = 72.
For powers of the same base, indices are added when multiplying and subtracted when dividing. The numerator is 3^(2−5) = 3⁻³ and the denominator is 3^(−1+0) = 3⁻¹, so the quotient is 3^(−3−(−1)) = 3⁻², which equals 1/9.
Let x = 0.4444…, so that 10x = 4.4444…. Subtracting the first equation from the second gives 9x = 4, and therefore x = 4/9, which is already in its lowest terms.
The digit in the thousands place is 3, so the deciding digit is the hundreds digit, which is 5. Because 5 means round up, the thousands digit increases from 3 to 4 and the last three digits become zeros, giving 4 874 000.
As decimals the values are −0.75, −0.667, −0.5 and 0.4. For negative numbers, the value furthest from zero is the smallest, so −3/4 comes first, then −2/3, then −1/2, and finally the positive decimal 0.4.
To change a fraction into a percentage, multiply it by 100. Here 5/8 × 100 = 500 ÷ 8 = 62.5, so the fraction equals 62.5%.
Repeated division by primes gives 360 = 2 × 180 = 2 × 2 × 90 = 2 × 2 × 2 × 45, and 45 = 3 × 3 × 5. Collecting repeated factors into powers gives 2³ × 3² × 5, which checks to 8 × 9 × 5 = 360.
The distance per litre is found by dividing: 180 ÷ 15 = 12 km per litre. Multiplying by 22 litres gives 22 × 12 = 264 km, so the car travels 264 km.
A negative index means take the reciprocal of the base, so (2/3)⁻² = (3/2)². Squaring numerator and denominator gives 9/4.
After a 15% reduction the sale price is 85% of the original price. Dividing by 0.85 gives 340 ÷ 0.85 = 400, and checking: 15% of 400 is 60, and 400 − 60 = 340.
When changing from pula to pounds, divide the amount in pula by the value of one pound. Here 4 300 ÷ 17.20 = 250, so the traveller receives £250.
A measurement correct to 1 decimal place can be up to 0.05 below the stated value, so the lower bounds are 8.35 cm and 5.55 cm. The smallest possible area is 8.35 × 5.55 = 46.3425 cm².
Compound interest gives an amount of 5 000 × 1.04² = 5 000 × 1.0816 = 5 408. The interest earned is the amount minus the principal, so 5 408 − 5 000 = P408.
Use the order of operations: multiplication and division are done before addition and subtraction. Here 3 × 4 = 12 and 8 ÷ 2 = 4, so the expression becomes 5 + 12 − 4 = 13.
A fraction can be converted to a decimal by dividing the numerator by the denominator. Since 7 ÷ 20 = 0.35, the decimal form of 7/20 is 0.35.
Write the prime factorisations: 48 = 2^4 × 3 and 72 = 2^3 × 3^2. The HCF uses the lower power of each common prime factor, so 2^3 × 3 = 24.
Multiply the coefficients: 4.5 × 2 = 9. Then add the indices of the powers of 10: 10^4 × 10^−2 = 10^2, so the answer is 9 × 10^2.
Simple interest is I = PRT/100. Substituting P = 2500, R = 6 and T = 3 gives I = (2500 × 6 × 3)/100 = 450, so the interest is P450.
Multiply each part by 10 to remove the decimals, giving 8 : 12 : 20. Divide each term by the common factor 4 to obtain 2 : 3 : 5.
Dividing by a fraction is the same as multiplying by its reciprocal. So 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.
At noon the temperature was −7 + 12 = 5°C. By midnight it had fallen 5°C, so it was 5 − 5 = 0°C.
The difference between the largest and smallest shares is 7 − 2 = 5 parts. Since 5 parts equal P450, one part is P90. The total is 2 + 3 + 7 = 12 parts, so 12 × 90 = P1 080.
First combine each bracket: 1/2 + 1/3 = 5/6 and 1/4 − 1/5 = 1/20. Then divide: 5/6 ÷ 1/20 = 5/6 × 20 = 100/6 = 50/3.
Each place value column is ten times the value of the column to its right. The digit 3 is in the hundred-thousands column, so its value is 3 × 100 000 = 300 000.
The lowest common denominator of 8 and 6 is 24. Rewrite the fractions as 9/24 and 4/24, then add the numerators to get 13/24.
List multiples of 12 and 18 until a common value appears: 12, 24, 36 and 18, 36. The first shared multiple is 36, so the LCM is 36.
The actual increase is 92 − 80 = 12. Percentage increase is found by dividing the increase by the original amount and multiplying by 100, giving 12/80 × 100 = 15%.
The ratio has 5 + 4 = 9 equal parts. One part is P720 ÷ 9 = P80, so the larger share is 5 × P80 = P400.
Divide the coefficients: 8.1 ÷ 2.7 = 3. For the powers of ten, subtract the indices: 10^6 ÷ 10^2 = 10^(6−2) = 10^4. The answer is therefore 3 × 10^4.
Let x = 0.222... . Then 10x = 2.222... . Subtracting gives 9x = 2, so x = 2/9.
Multiply 6 by 35 to get 210. There are three decimal places in total in 0.6 and 0.35, so place the decimal point three places from the right, giving 0.210 = 0.21.
Write 84 = 2^2 × 3 × 7. For a perfect square, every prime exponent must be even, so n must supply one factor of 3 and one factor of 7. Thus the smallest n is 3 × 7 = 21.
Write 16 as 2^4, so 16^(3/4) = (2^4)^(3/4) = 2^3 = 8. Since 2^(−3) = 1/8, the product is 8 × 1/8 = 1.
Like terms contain exactly the same variables with the same powers, so they can be combined by adding or subtracting their coefficients. Here 5a + 3a gives 8a, and -2b - b gives -3b, producing 8a - 3b.
Divide both sides of the equation by 3 to get x - 2 = 6, then add 2 to both sides. This gives x = 8; substituting 8 back into the original equation confirms 3(6) = 18.
Use the distributive law: multiply each term inside the bracket by 4. Thus 4 × 2x = 8x and 4 × (-3) = -12, giving 8x - 12.
The highest common factor of 6x^2 and 9x is 3x, because 3x divides both terms. Taking 3x out leaves 2x - 3 inside the bracket, so the fully factorised expression is 3x(2x - 3).
Subtract 5 from both sides to get -2x < 6. Dividing by -2 reverses the inequality sign, so x > -3.
Adding the two equations eliminates y, because y + (-y) = 0, giving 3x = 12, so x = 4. Substituting x = 4 into x - y = 1 gives y = 3, and the solution checks in both equations.
Recognise the numerator as a difference of two squares: x^2 - 9 = (x - 3)(x + 3). Cancelling the common factor x + 3 leaves x - 3, provided x is not -3.
(2x^2)^3 means 2^3 × (x^2)^3. This equals 8x^6 because 2 cubed is 8 and powers are multiplied when raising a power to a power.
Let the width be w cm; then the length is w + 3 cm. The perimeter is 2(w + w + 3) = 34, so 4w + 6 = 34, giving w = 7. The dimensions 7 cm by 10 cm give a perimeter of 34 cm.
Factorise the quadratic as (x - 2)(x - 3) = 0, since -2 and -3 multiply to 6 and add to -5. Setting each factor equal to zero gives x = 2 or x = 3.