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Write both fractions over the lowest common denominator, 12: 3/4 = 9/12 and 5/6 = 10/12. Adding gives 19/12, and since 19 ÷ 12 = 1 remainder 7, the answer is 1 7/12.
Subtracting a negative number is the same as adding, so −(−3) becomes +3. The calculation is then −5 + 8 + 3 = 6.
Dividing 84 repeatedly by its smallest prime factors gives 84 = 2 × 42 = 2 × 2 × 21 = 2² × 3 × 7. Every factor in the final product must be prime, and 2² × 3 × 7 checks to 84.
Multiply the number parts: 4.2 × 3 = 12.6. Add the indices: 10⁵ × 10⁻² = 10³, so the result is 12.6 × 10³. For standard form the first part must lie between 1 and 10, so rewrite this as 1.26 × 10⁴.
The sale price represents 90% of the original price, so the original price multiplied by 0.9 equals £54. Dividing gives 54 ÷ 0.9 = £60, and a check of 10% of £60 (£6) confirms £60 − £6 = £54.
The ratio 3 : 4 contains 3 + 4 = 7 equal parts, so one part is £420 ÷ 7 = £60. The larger share is 4 parts, which is 4 × £60 = £240.
Rounding to 1 decimal place means numbers are rounded to the nearest 0.1, so the interval of possible values has width 0.1. Half of this is 0.05, and the lower bound is 4.7 − 0.05 = 4.65, which is included because 4.65 rounds up to 4.7.
Let x = 0.4444…, so 10x = 4.4444…. Subtracting gives 10x − x = 9x = 4, and therefore x = 4/9, which is already in its simplest form.
For compound interest, multiply by the multiplier 1.03 once for each year: 5000 × 1.03² = 5000 × 1.0609 = £5304.50. The second year's interest is calculated on £5150, which includes the first year's interest.
Each measurement has a half-interval of 0.05 cm, so the upper bounds are 8.45 cm and 5.25 cm. To maximise the area, multiply the upper bounds together: 8.45 × 5.25 = 44.3625 cm².
Standard form is written as A × 10ⁿ where 1 ≤ A < 10. Moving the decimal point in 0.000305 four places to the right gives 3.05, so the power of ten is −4.
Write each number as a product of prime factors: 36 = 2² × 3² and 60 = 2² × 3 × 5. The HCF uses the lowest power of each common prime factor, giving 2² × 3 = 12.
To convert a fraction to a percentage, multiply by 100. So 7/20 × 100 = 7 × 5 = 35, giving 35%.
First multiply the two negative numbers: (−4) × (−6) = 24. Then divide by −3, giving 24 ÷ (−3) = −8.
The increase is £300 − £250 = £50. Percentage increase = increase ÷ original × 100, so 50 ÷ 250 × 100 = 20%.
The 5 parts representing flour are worth 350 g, so 1 part is 350 ÷ 5 = 70 g. Sugar is 2 parts, so the mass is 2 × 70 = 140 g.
Write both numbers with the same power of ten: 3.8 × 10⁵ = 0.38 × 10⁶. Then 5.2 × 10⁶ + 0.38 × 10⁶ = 5.58 × 10⁶.
The lower bound of x is 4.65 and the upper bound of y is 2.35. To make x − y as small as possible, subtract the largest possible y from the smallest possible x, giving 4.65 − 2.35 = 2.30.
The denominator 3 means take the cube root, and 27^(1/3) = 3. The numerator 2 means square the result, so 3² = 9.
The bounds for 8.4 correct to 2 significant figures are 8.35 g to 8.45 g, so the maximum absolute error is 0.05 g. Percentage error = 0.05 ÷ 8.4 × 100 = 0.5952…%, which is 0.60% to 2 significant figures.
The LCM is the smallest positive integer that is a multiple of both numbers. Prime factorising gives 12 = 2² × 3 and 18 = 2 × 3², so the LCM uses the highest power of each prime: 2² × 3² = 36. 6 is the HCF, and 72 is a common multiple but not the lowest.
0.375 means 375/1000, which simplifies by dividing numerator and denominator by 125 to give 3/8. Alternatively, 3 ÷ 8 = 0.375. The other fractions give different decimal values.
Multiplying by 10³ moves the decimal point three places to the right. 4.05 × 1000 = 4050, so the ordinary number is 4050.
The decrease is £12,000 − £9,600 = £2,400. Percentage decrease = decrease ÷ original × 100 = 2400 ÷ 12000 × 100 = 20%. Dividing the new value by the original gives 80%, which is the value remaining, not the decrease.
Total number of ratio parts = 2 + 3 + 4 = 9. One part is £720 ÷ 9 = £80, so the smallest share (2 parts) is 2 × £80 = £160. The £80 option is one part, not the smallest share.
Rounding to 1 significant figure gives 4.8 ≈ 5, 21.2 ≈ 20 and 0.49 ≈ 0.5. Then (5 × 20) ÷ 0.5 = 100 ÷ 0.5 = 200. Dividing by 0.5 doubles the numerator.
When multiplying powers with the same base, add the indices: 3⁴ × 3⁵ = 3⁹. When dividing, subtract the indices: 3⁹ ÷ 3⁶ = 3^(9−6) = 3³.
Look for the largest square factor of 50, which is 25. Since √50 = √(25 × 2) = √25 × √2 = 5√2, the simplified surd is 5√2. 50 has no square factor larger than 25.
Use multipliers: a 12% increase is ×1.12 and a 12% decrease is ×0.88. Overall multiplier = 1.12 × 0.88 = 0.9856, which is a 1 − 0.9856 = 0.0144 = 1.44% decrease. The changes do not cancel because the decrease is calculated on the larger amount.
Multiply numerator and denominator by the conjugate √5 + 2. The denominator becomes (√5)² − 2² = 5 − 4 = 1, so the fraction simplifies to √5 + 2, i.e. 2 + 1√5.
The first significant figure is the 4 in the thousandths place. The second is the 7; the next digit is 0, so the 7 stays unchanged and trailing zeros are not needed.
Indices are evaluated first, so 3² = 9. Then multiplication: 2 × 9 = 18, and finally addition: 5 + 18 = 23.
A prime number has exactly two factors, 1 and itself. 31 is not divisible by 2, 3 or 5, and √31 is less than 6, so it has no other factor.
Dividing 84 by 2 gives 42, by 2 again gives 21, and 21 = 3 × 7. All factors in 2² × 3 × 7 are prime.
Let x = 0.444...; then 10x = 4.444..., so 9x = 4 and x = 4/9. This fraction is already in simplest form.
Each year the amount is multiplied by 1.03. After two years, £2000 × 1.03² = £2000 × 1.0609 = £2121.80.
The sale price is 80% of the original price because 100% − 20% = 80%. Divide £64 by 0.8 to get £80.
This is the difference of two squares: (2)² − (√3)² = 4 − 3 = 1.
Write 27 as 3³, so 27^(x − 1) = 3^(3x − 3). Equating the powers gives 2x = 3x − 3, so x = 3.
For the largest density, use the upper bound of mass and the lower bound of volume. Mass upper bound = 255 g and volume lower bound = 12.5 cm³, so density ≤ 255 ÷ 12.5 = 20.4 g/cm³.
Subtracting a negative number is the same as adding its positive value, so the expression becomes -7 + 4 + 3. Working from left to right gives -3 + 3, which equals 0.
Use a common denominator of 12: 2/3 = 8/12 and 1/4 = 3/12. Adding the numerators gives 8/12 + 3/12 = 11/12, which is already in simplest form.
15% can be split into 10% plus 5%. Ten percent of £80 is £8, and 5% is half of that, £4, so the total is £8 + £4 = £12.
Multiply the coefficients: 3.2 × 2.5 = 8.0. Add the powers of 10: 4 + (-2) = 2, so the result is 8.0 × 10², which is 8 × 10² in standard form.
Simple interest is the same each year: 4% of £1500 is £60. Over 3 years the total interest is 3 × £60 = £180.
The difference in the ratio is 7 - 5 = 2 parts, and this equals £84, so one part is £42. Alex receives 5 parts, which is 5 × £42 = £210.
When a number is rounded to 1 decimal place, the lower bound is half of 0.1 below the rounded value. Half of 0.1 is 0.05, so the lower bound is 7.3 - 0.05 = 7.25.
A fractional index of 3/4 means take the fourth root and then cube the result. The fourth root of 16 is 2, and 2³ = 8.
Divide the coefficients: 6.4 ÷ 8 = 0.8. Subtract the powers: 5 - (-3) = 8, giving 0.8 × 10⁸. To write this in standard form, adjust to 8 × 10⁷.
Convert both mixed numbers to improper fractions: 2 1/3 = 7/3 and 1 1/2 = 3/2. Multiply to get 21/6, which simplifies to 7/2 or 3 1/2.