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In 4,372 the digits occupy thousands, hundreds, tens and units places from the left. The 7 sits in the tens place, so its value is 7 × 10 = 70, not 7 or 700.
Compare tenths first: 0.6, 0.65 and 0.605 all have 6 tenths, while 0.06 has 0 tenths and is smallest. Among the rest compare hundredths: 0.65 has 5 hundredths, more than the 0 hundredths of 0.6 and 0.605, so 0.65 is largest.
Rewrite 3/4 as 6/8 so both fractions share the denominator 8, then add the numerators: 6/8 + 1/8 = 7/8. This fraction is already in simplest form because 7 is prime and does not divide 8.
A prime has exactly two factors, 1 and itself. The number 29 is not divisible by 2, 3 or 5, and since √29 is about 5.4 no larger factor needs testing, so 29 is prime.
A 25% reduction means the buyer pays 75% of the original price. Compute 0.75 × 48 = 36, so the sale price is £36; the £12 is the discount, not the price.
Write 36 = 2² × 3² and 60 = 2² × 3 × 5. The HCF takes the lowest power of each shared prime: 2² × 3 = 12.
When multiplying powers with the same base, add the indices: 2³ × 2⁴ = 2^(3+4) = 2⁷. Multiplying the indices or the bases gives incorrect results.
Convert everything to decimals: 0.4 = 0.4, 1/3 ≈ 0.333, 40% = 0.4, and 0.35 = 0.35. The smallest is 0.35, then 1/3 ≈ 0.333... wait, comparing carefully 0.35 > 1/3, so the order is 1/3, 0.35, 0.4 = 40%.
Each ratio part equals 350 ÷ 5 = 70 g. Sugar is 2 parts, so 2 × 70 = 140 g.
Standard form is a × 10ⁿ with 1 ≤ a < 10. Moving the decimal point 4 places right gives 3.6, so the index is −4: 3.6 × 10⁻⁴.
A 15% increase means the new amount is 115% of the original. Compute 240 × 1.15 = 276, so the increased amount is £276.
A fraction in simplest form terminates only if its denominator has no prime factors other than 2 and 5. Since 8 = 2³, 7/8 = 0.875 terminates.
Dividing by 2/3 is the same as multiplying by its reciprocal 3/2. So 5/6 × 3/2 = 15/12 = 5/4.
Evaluate each prime power: 2² = 4, 3 = 3, 5² = 25. Multiplying gives 4 × 3 × 25 = 12 × 25 = 300.
The first two significant figures of 472 are 4 and 7. The next digit is 2, which is less than 5, so the 7 stays and the units digit becomes 0, giving 470.
Write 12 = 2² × 3 and 18 = 2 × 3². The LCM takes the highest power of each prime: 2² × 3² = 4 × 9 = 36.
Squaring a fraction means squaring numerator and denominator separately: (3/4)² = 3²/4² = 9/16. This is already in simplest form.
Evaluate each: 3² = 9, 2³ = 8, 2⁴ = 16, and 4¹ = 4. The largest is 16, so 2⁴ has the greatest value.
Doubling each hour multiplies the count by 2 once per hour, so after 6 hours the count is 500 × 2⁶. Repeated multiplication by a constant is modelled by a power.
Convert the time to hours: 2 hours 30 minutes = 2.5 hours. Speed = distance ÷ time = 150 ÷ 2.5 = 60 km/h.
Let x = 0.121212... Then 100x = 12.121212..., and subtracting gives 99x = 12, so x = 12/99. This simplifies to 4/33, which has the same value.
The first four primes are 2, 3, 5 and 7. Their product is 2 × 3 × 5 × 7 = 210.
Repeatedly doubling from 2: 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64. Therefore n = 6.
Use multipliers: ×1.20 then ×0.80 gives 1.20 × 0.80 = 0.96. The final amount is 96% of the original, an overall decrease of 4%.
If N = p^a × q^b × r^c, the number of factors is (a+1)(b+1)(c+1). Here the indices are 4, 2 and 1, so the count is 5 × 3 × 2 = 30.
An 8% increase means the result is 108% of the original number. Dividing 540 by 1.08 gives 500, and 500 × 1.08 = 540 confirms the answer.
An irrational number cannot be written as a fraction of two integers. √2 has a non-terminating, non-recurring decimal expansion, so it is irrational. The others are rational: √49 = 7, 0.75 = 3/4 and 22/7 is a fraction.
A negative index means take the reciprocal of the positive power: 5⁻² = 1/5² = 1/25. The negative sign belongs to the exponent, not to the value of the result.
Work out each power first: 2³ = 8 and 3² = 9. Then 8 × 9 × 7 = 72 × 7 = 504.
A multiple of 6 must be divisible by both 2 and 3. 18 = 6 × 3, but 18 ÷ 4 = 4.5, so it is not a multiple of 4. The other options are all divisible by 4 as well as by 6.
Let x = 0.363636... Then 100x = 36.363636..., so 100x − x = 36, giving 99x = 36 and x = 36/99 = 4/11. Checking, 4 ÷ 11 = 0.363636..., as required.
To find a fraction of a fraction, multiply: (3/5) × (2/9) = 6/45. Simplifying by dividing top and bottom by 3 gives 2/15.
Rounding to one decimal place means values from 4.65 up to (but not including) 4.75 round to 4.7. The lower bound is exactly 4.65, since 4.65 rounds up to 4.7.
The price including VAT is 120% of the pre-VAT price. Dividing £48 by 1.2 gives £40, and £40 × 1.2 = £48 confirms it.
A negative index means take the reciprocal: (2/3)⁻¹ = 1 ÷ (2/3) = 3/2. The result is greater than 1 because the original fraction is less than 1.
Standard form is A × 10ⁿ where 1 ≤ A < 10 and n is an integer. Only 3.2 × 10⁵ has a first factor between 1 and 10 and a power of 10 as the second factor.
Multiply: 8000 × 2.5 × 10⁻⁵ = 8 × 10³ × 2.5 × 10⁻⁵ = 20 × 10⁻² = 2 × 10⁻¹ kg. This equals 0.2 kg, but the answer must be given in standard form.
Take the square root of each factor: √1.44 = 1.2 and √(10⁶) = 10³. So the answer is 1.2 × 10³, which equals 1200. Checking, 1200² = 1 440 000 = 1.44 × 10⁶.
We need 1.05ⁿ > 1.5. Testing powers: 1.05⁸ ≈ 1.477 and 1.05⁹ ≈ 1.551. So the investment first exceeds 1.5 times the original after 9 complete years.
The exponents are 2 (on 2), 3 (on 3) and 1 (on 5). Their sum is 2 + 3 + 1 = 6. Checking, 4 × 27 × 5 = 540, so the factorisation is correct.
Subtract 3x from both sides to get 2x − 7 = 9, then add 7 to obtain 2x = 16, so x = 8. Substituting back gives 33 = 33, confirming the solution.
The expression is a difference of two squares, a² − b² = (a − b)(a + b), with a = x and b = 7. Expanding (x − 7)(x + 7) gives x² − 49, which matches the original polynomial.
Slope equals the change in y divided by the change in x: (13 − 5)/(6 − 2) = 8/4 = 2. The line rises two units vertically for every one unit horizontally.
Multiply the coefficients 3 and 4 to get 12, then add exponents of like bases: a² · a = a³ and b · b³ = b⁴. The product is therefore 12a³b⁴.
Substitute x = −2: 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15. The square makes the first term positive and the negative coefficient times a negative input makes the middle term positive.
Subtract 4 from both sides to get −2x > 6, then divide by −2 and reverse the inequality sign, giving x < −3. Reversing the sign is required whenever an inequality is divided by a negative number.
The nth term of an arithmetic sequence is a + (n − 1)d, so the 10th term is 7 + 9(4) = 7 + 36 = 43. The first term already counts as term one, so only nine differences are added.
The absolute value equation splits into x − 3 = 7 or x − 3 = −7. Solving each gives x = 10 and x = −4, and both check since |7| = 7 and |−7| = 7.
The discriminant is b² − 4ac = 36 − 36 = 0, and a zero discriminant means exactly one repeated real root. Factoring gives (x − 3)² = 0, so the root is x = 3 with multiplicity two.
Here a = 2, b = 5, c = −3, so the discriminant is 25 + 24 = 49 and √49 = 7. The roots are (−5 ± 7)/4, giving 2/4 = 1/2 and −12/4 = −3.