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To round to the nearest hundred, look at the digit in the tens column, which is 3. Because 3 is less than 5, the hundreds digit (7) stays the same and the digits after it become zeros. This gives 4,700.
Order of operations (BEMDAS) requires multiplication before addition. So 2 × 5 = 10 first, then 6 + 10 = 16.
Two fractions are equivalent when the numerator and denominator are both multiplied (or divided) by the same number. Multiplying 3 and 4 by 3 gives 9/12, and cross-multiplying confirms 3 × 12 = 4 × 9 = 36.
First find 15% of €80 by working out 0.15 × 80 = 12, which is the amount of the reduction. Subtracting the discount from the original price gives 80 − 12 = €68.
Subtracting a negative number is the same as adding its positive counterpart, so the calculation becomes −3 + 7. Starting at −3 and moving 7 places to the right on the number line lands on 4.
Write both numbers as products of primes: 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3. The HCF is the product of the prime factors common to both lists, which is 2 × 2 × 3 = 12.
Standard form is written as a × 10ⁿ where a is at least 1 and less than 10. Moving the decimal point four places to the right in 0.00047 gives 4.7, so the power of ten must be negative four.
Add the parts of the ratio: 3 + 5 = 8, so the money is split into 8 equal shares. One share is 240 ÷ 8 = €30, and Ben's 5 shares come to 5 × 30 = €150.
Let x = 0.444… Then 10x = 4.444… Subtracting the first equation from the second gives 9x = 4, so x = 4/9. The fraction cannot be simplified further because 4 and 9 share no common factor.
When multiplying powers with the same base you add the indices, so 2³ × 2⁵ = 2⁸. When dividing you subtract the indices, so 2⁸ ÷ 2⁴ = 2⁽⁸⁻⁴⁾ = 2⁴, which equals 16.
The place values in 4,706 are thousands, hundreds, tens and units from left to right. The digit 7 sits in the hundreds column, so it represents 7 hundreds, which is 700.
Multiplying two negative numbers gives a positive result. Since 4 × 6 = 24, it follows that (−4) × (−6) = 24.
To find three fifths of 45, first divide 45 by the denominator 5 to get 9. Then multiply by the numerator 3: 9 × 3 = 27.
The LCM is the smallest number that both 12 and 18 divide into exactly. Listing multiples of 12 gives 12, 24, 36, and multiples of 18 give 18, 36, so the first shared multiple is 36.
Write both fractions with the common denominator 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.
The first three significant figures are 3, 6 and 7. The next digit is 8, so the third figure 7 rounds up to 8, giving 3.68.
The difference in the ratio is 7 − 2 = 5 parts, and this equals €45, so one part is €9. The total is 2 + 7 = 9 parts, giving 9 × €9 = €81.
The new price represents 120% of the original price, because 100% + 20% = 120%. Dividing €360 by 120 gives €3 per 1%, so the original 100% is €300.
After a 20% reduction the price is 80% of the original. The second reduction leaves 90% of that, and 0.8 × 0.9 = 0.72, so the final price is 72% of the original, a total reduction of 28%.
Multiply the coefficients: 3 × 4 = 12. Add the indices: 5 + (−2) = 3, giving 12 × 10³. Standard form requires a coefficient between 1 and 10, so 12 × 10³ is rewritten as 1.2 × 10⁴.
Ascending means from smallest to largest. Comparing the decimals place by place gives 0.0006 < 0.006 < 0.06 < 0.6, so only the second list follows the correct order.
A prime number has exactly two factors: 1 and itself. 29 cannot be divided exactly by any whole number other than 1 and 29, whereas 21, 27 and 33 are all divisible by 3.
A multiple of 6 must be divisible by both 2 and 3. 42 is even and its digits add to 6, which is divisible by 3, so 42 is a multiple of 6 because 6 × 7 = 42.
Dividing by 4/9 is the same as multiplying by its reciprocal, 9/4. So 2/3 × 9/4 = 18/12, which simplifies to 3/2.
The ratio 5 : 2 means each part is 350 ÷ 5 = 70 g. Sugar is 2 parts, so 2 × 70 = 140 g.
Multiply 6 by 4 to get 24. There are two decimal places in 0.06 and one in 0.4, giving three decimal places in total, so the answer is 0.024.
Average speed is distance divided by time. 84 ÷ 1.5 = 56, so the average speed is 56 km/h.
The index −3 means move the decimal point three places to the left. Starting from 5.06 gives 0.00506.
With compound interest the value after 2 years is P × 1.04². Since 1.04² = 1.0816, P = 1081.60 ÷ 1.0816 = 1000.
In a perfect square every prime index must be even. Here 2 has index 3 and 5 has index 1, so multiplying by 2 × 5 = 10 gives 2⁴ × 3² × 5², a perfect square.
The factors of 24 include 1, 2, 3, 4, 6, 8, 12 and 24. The factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18 and 36. The largest number in both lists is 12, so the HCF is 12.
First divide: (−18) ÷ (−3) = 6. Then multiply: 4 × (−2) = −8. Finally add: 6 + (−8) = −2.
0.36 means 36 hundredths, so it can be written as 36/100. Dividing the numerator and denominator by 4 gives 9/25, which cannot be simplified further.
To add numbers in standard form, first write them with the same power of 10. 5.2 × 10^6 = 52 × 10^5, so the sum is (52 + 3.4) × 10^5 = 55.4 × 10^5. Adjusting to standard form gives 5.54 × 10^6.
Like terms in x are combined by adding and subtracting their coefficients: 3 + 5 − 2 = 6, so the expression simplifies to 6x. The power of x does not change because the terms are being added and subtracted, not multiplied together.
Subtract 7 from both sides to isolate the x-term, giving 2x = 12. Divide both sides by the coefficient 2 to get x = 6, and checking: 2(6) + 7 = 19, which is correct.
The 4 outside the bracket multiplies every term inside it: 4 × 2x = 8x and 4 × (−3) = −12. Putting the two products together gives 8x − 12, and the sign of the second term stays negative.
Collect the x-terms on one side and the numbers on the other: 5x − 2x = 12 + 3, which gives 3x = 15. Dividing both sides by 3 gives x = 5, and substituting back gives 22 on each side.
Two numbers are needed that multiply to give 12 and add to give 7; those numbers are 3 and 4. So x² + 7x + 12 = (x + 3)(x + 4), and expanding this back out returns the original expression.
Adding the two equations eliminates y, since +y and −y cancel, giving 3x = 12 and so x = 4. Substituting into x − y = 1 gives 4 − y = 1, so y = 3, and checking in 2x + y = 11 gives 8 + 3 = 11.
Add 5 to both sides to get 3x ≤ 15, then divide both sides by 3. Because 3 is positive, the inequality sign keeps pointing the same way, giving x ≤ 5; testing x = 0 gives −5 ≤ 10, which is true.
Subtract 2l from both sides to get P − 2l = 2w, then divide every term by 2, which gives w = (P − 2l)/2 = P/2 − l. Checking with P = 20 and l = 6 gives w = 10 − 6 = 4, and 2(6) + 2(4) = 20 as expected.
Two numbers that multiply to −14 and add to −5 are −7 and 2, so the equation factorises as (x − 7)(x + 2) = 0. Setting each bracket equal to zero gives x = 7 or x = −2, and checking 7² − 5(7) − 14 = 49 − 35 − 14 = 0 confirms the first root.
The numerator is a difference of two squares, so x² − 9 = (x − 3)(x + 3). The common factor (x + 3) cancels with the denominator, leaving x − 3; testing x = 1 gives (1 − 9)/4 = −2, which equals 1 − 3.
When a = −2, a² = 4, so 2a² = 8; also 3b = 12. Therefore 2a² − 3b = 8 − 12 = −4.
Multiply the coefficients: 4 × 3 = 12. For the powers of x, add the indices: x³ × x² = x³⁺² = x⁵, so the result is 12x⁵.
Divide both sides by 3 to get x − 4 = 6. Then add 4 to both sides, giving x = 10.
Use the four products: x·x = x², x·(−2) = −2x, 5·x = 5x, and 5·(−2) = −10. Combining −2x and 5x gives 3x, so the expansion is x² + 3x − 10.
The highest common factor of 6x² and 9x is 3x. Dividing each term by 3x gives 2x and −3, so the fully factorised form is 3x(2x − 3).
Substitute x = y + 2 into 3x + 2y = 16: 3(y + 2) + 2y = 16, so 5y + 6 = 16 and y = 2. Then x = y + 2 = 4.