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A 25% discount removes one quarter of the price, and one quarter of $80 is $20. Subtracting gives $80 − $20 = $60. Equivalently, the buyer pays 75% of $80, which is 0.75 × 80 = $60.
Multiplication and division are done before addition and subtraction. So 4 × 2 = 8 and 6 ÷ 3 = 2, giving 3 + 8 − 2 = 9. Working strictly left to right would break the order-of-operations convention.
Compare the tenths digit first: 0.8 has 8 tenths, while the others have 7, 6 and 0 tenths. Since 8 tenths is the greatest, 0.8 is the largest value even though 0.099 has more digits.
Collect like terms separately: 5a − 2a = 3a and 3b + b = 4b. The simplified expression is therefore 3a + 4b. Terms in a and terms in b cannot be combined into one term.
The ratio flour:milk is 2:3, so flour is two thirds of the milk quantity. Two thirds of 12 is 8. Checking: 8:12 simplifies to 2:3, which matches the recipe.
Convert to decimals: 2/5 = 0.4 and 1/2 = 0.5. The values are then 0.35, 0.4, 0.5, 0.6, which ascend correctly. Any list that places 0.6 before 0.5 or 0.4 before 0.35 is out of order.
Each term increases by a constant difference of 5, since 9 − 4 = 5 and 14 − 9 = 5. Adding 5 to 19 gives 24. This is an arithmetic sequence with common difference 5.
Average speed is distance divided by time, so 150 km ÷ 2 h = 75 km/h. Multiplying distance by time would give the wrong units, not a speed.
Add 7 to both sides to get 4x = 28, then divide both sides by 4 to get x = 7. Substituting back: 4 × 7 − 7 = 21, which confirms the solution.
The GST-inclusive price equals 1.15 times the pre-GST price, so divide $45 by 1.15 to get about $39.13. Taking 15% of $45 and subtracting would be wrong because the 15% applies to the smaller pre-GST amount.
Multiply each term: x·x = x², x·(−2) = −2x, 5·x = 5x, and 5·(−2) = −10. Combining −2x + 5x gives +3x, so the result is x² + 3x − 10.
Dividing 84 by 2 twice gives 21, and 21 = 3 × 7, so 84 = 2² × 3 × 7. A prime factorisation must use only prime numbers, and 2, 3 and 7 are all prime.
After one year the balance is 500 × 1.04 = $520. After the second year it is 520 × 1.04 = $540.80, because interest is earned on the interest already added. Compound growth uses the factor 1.04² = 1.0816.
Gradient equals the change in y divided by the change in x: (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2. The line rises 8 units over a run of 4 units.
The first red has probability 5/8. With one red removed, 4 of the remaining 7 counters are red, so the second is 4/7. Multiplying gives (5/8) × (4/7) = 20/56 = 5/14.
Subtract 4 from both sides to get 3x < 15, then divide by the positive number 3, which keeps the inequality direction: x < 5. Dividing by a negative number would be the only case that reverses the sign.
When multiplying powers with the same base, add the exponents: 2⁵ × 2³ = 2^(5+3) = 2⁸. This equals 256, which matches 32 × 8.
Let the width be w, so the length is w + 3 and w(w + 3) = 40. The equation w² + 3w − 40 = 0 factorises as (w + 8)(w − 5) = 0, giving w = 5 since width must be positive. Checking: 5 × 8 = 40.
The increase is $15, and percentage change is measured against the original value: 15 ÷ 60 = 0.25, or 25%. Measuring against the new price instead would give a different, incorrect figure.
The gradient is (10 − 2) ÷ (5 − 1) = 8 ÷ 4 = 2. Using the point (1, 2): 2 = 2(1) + c gives c = 0, so the equation is y = 2x. The point (5, 10) also satisfies y = 2x.
Rearranging gives x² − 2x − 15 = 0, which factorises as (x − 5)(x + 3) = 0. The solutions are x = 5 and x = −3. Checking x = −3: 9 = −6 + 15, which holds.
Multiply the coefficients: 3 × 4 = 12. Add the exponents of like bases: x² × x = x³ and y × y³ = y⁴. The product is 12x³y⁴.
Simple interest is I = P × r × t, so 360 = 2000 × 0.06 × t = 120t. Dividing gives t = 3 years. Checking: 2000 × 0.06 × 3 = $360.
The volume of a cone is one third of the volume of a cylinder with the same base and height, since V = ⅓πr²h. One third of 90 cm³ is 30 cm³. This fixed ratio holds regardless of the actual radius and height.
Factor out 2: 2(x² + 6x) + 10. Completing the square inside gives 2[(x + 3)² − 9] + 10 = 2(x + 3)² − 18 + 10 = 2(x + 3)² − 8, so k = −8. The constant must be adjusted after completing the square.
Multiplying the unknown n by 6 gives 6n, and adding 9 to that product gives 6n + 9. Setting this equal to the result 45 produces the equation 6n + 9 = 45. The order of operations matters, so the 9 is added after the multiplication, not before.
The cost has a fixed part of $3 that is charged regardless of distance, plus $2 multiplied by the number of kilometres k. So the total is C = 2k + 3. The variable part is the per-kilometre charge, and the constant is the flagfall.
Following the order of operations, evaluate the power first: 2² = 4. Then do the multiplication 3 × 4 = 12, giving 5 + 12 − 4. Working left to right gives 17 − 4 = 13.
The ratio 5 : 2 means each part of flour corresponds to a fixed number of grams. Since 5 parts equal 350 g, one part is 350 ÷ 5 = 70 g. Sugar is 2 parts, so 2 × 70 = 140 g.
A common factor must divide both numbers exactly. The number 12 divides 36 (36 ÷ 12 = 3) and also divides 60 (60 ÷ 12 = 5), so it is a common factor. Checking each option by division confirms only 12 works for both.
Adding a positive deposit to a negative balance means moving right on the number line: −40 + 65. Since 65 is larger than 40, the result is positive, and 65 − 40 = 25. So the new balance is $25.
This is an arithmetic sequence with first term 4 and common difference 3. The nth term is 4 + 3(n − 1), so for n = 8 we get 4 + 3 × 7 = 4 + 21 = 25.
The decimal 0.35 means 35 hundredths, so it equals 35/100. Dividing the numerator and denominator by their common factor 5 gives 7/20, which is in simplest form because 7 and 20 share no factor other than 1.
A 20% discount means the sale price is 80% of the original price. If 80% equals $96, then 1% equals $1.20 and 100% equals $120. So the original price was $120.
Expand each bracket: 5(2x − 3) = 10x − 15 and −4(x − 2) = −4x + 8, because the negative sign reverses the second term. Combining like terms gives 10x − 4x = 6x and −15 + 8 = −7, so the result is 6x − 7.
When a power is raised to a power, multiply the indices: (x³)² = x⁶. When multiplying powers of the same base, add the indices: x⁶ × x⁴ = x¹⁰.
The line has y-intercept 5 and gradient −2, so its equation is y = −2x + 5. Substituting x = 4 gives y = −8 + 5 = −3.
Subtract 7 from both sides to get −2x = −6. Dividing both sides by −2 gives x = 3. Substituting back checks: 7 − 2(3) = 7 − 6 = 1.
Increasing a number by 30% gives 130% of the original, so 1.3 times the original equals 91. Dividing 91 by 1.3 gives 70. Checking: 30% of 70 is 21, and 70 + 21 = 91.
Divide both sides by 2 to get P/2 = l + w, then subtract w from both sides to get l = P/2 − w. This isolates l correctly by undoing the operations in reverse order.
Set up a proportion: 8 L per 100 km means 0.08 L per km. For 250 km, the fuel needed is 250 × 0.08 = 20 L. Alternatively, 250 km is 2.5 times 100 km, so 2.5 × 8 = 20 L.
The numbers greater than 4 on a die are 5 and 6, so there are 2 favourable outcomes out of 6 equally likely outcomes. The probability is 2/6, which simplifies to 1/3.
Set 3n − 1 = 26, then add 1 to both sides to get 3n = 27, so n = 9. The 9th term is 3(9) − 1 = 26, which confirms the answer.
The 15% decrease is calculated on $40, giving a reduction of $6 and a price of $34. The 15% increase is then calculated on $34, adding $5.10 to give $39.10. Because the increase applies to a smaller amount, the final price is less than $40.
Apply the power to each factor: 2³ = 8, (x²)³ = x⁶ and (y³)³ = y⁹. Combining these gives 8x⁶y⁹. The coefficient 2 must also be cubed, not just multiplied by 3.
Factorising gives x² − 4x + 3 = (x − 1)(x − 3), so the intercepts are at x = 1 and x = 3. Their sum is 1 + 3 = 4, which also equals the negative of the coefficient of x.
In 40 minutes (2/3 hour) at 18 km/h the cyclist covers 12 km, and in 20 minutes (1/3 hour) at 12 km/h covers 4 km. Total distance is 16 km over 1 hour, so the average speed is 16 km/h.
Let the smaller integer be n, so the next is n + 1. Then (n + 1)² − n² = 2n + 1 = 27, giving 2n = 26 and n = 13. Checking: 14² − 13² = 196 − 169 = 27.
The given line has gradient 3, so a perpendicular line has gradient −1/3 because the gradients multiply to −1. Using y − 5 = −(1/3)(x − 2) and rearranging gives y = −(1/3)x + 17/3.
The difference between C and A is 5 − 3 = 2 parts, and this equals $45, so one part is $22.50. The total is 3 + 4 + 5 = 12 parts, giving 12 × $22.50 = $270.