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A 15% discount removes 0.15 × 180 = $27 from the price, so the sale price is 180 − 27 = $153. Equivalently, the customer pays 85% of the original price, and 0.85 × 180 = $153.
Standard form writes a number as a × 10^n where 1 ≤ a < 10 and n is an integer. Only 3.6 × 10^4 has the leading factor between 1 and 10, so it is the correct standard form.
Collect like terms separately: 5a − 2a = 3a and 3b + 4b = 7b. Combining these gives 3a + 7b, and the two types of term cannot be merged further because a and b are different variables.
When multiplying powers with the same base, add the indices: 2^3 × 2^4 = 2^(3+4) = 2^7, which equals 128. The base stays the same and only the exponents are added.
Pythagoras' theorem gives hypotenuse² = 6² + 8² = 36 + 64 = 100, so the hypotenuse is √100 = 10 cm. The result is a whole number because 6, 8 and 10 form a Pythagorean triple.
The mean is the total of the values divided by how many there are: (4 + 7 + 7 + 9 + 13) ÷ 5 = 40 ÷ 5 = 8. The sum is 40 and there are five data values.
An irrational number cannot be written as a fraction of two integers. Since 7 is not a perfect square, √7 has a non-terminating, non-repeating decimal expansion and is irrational.
Add 5 to both sides to get 3x = 21, then divide both sides by 3 to get x = 7. Substituting back gives 3(7) − 5 = 16, which confirms the solution.
There are 5 + 3 + 2 = 10 counters in total and 3 of them are blue, so the probability is 3/10. Probability is the number of favourable outcomes divided by the total number of equally likely outcomes.
Expanding gives x² − 3x + 4x − 12, and combining the middle terms −3x + 4x gives +x. The simplified result is therefore x² + x − 12.
Gradient equals the change in y divided by the change in x: (13 − 5) ÷ (6 − 2) = 8 ÷ 4 = 2. The line rises 8 units for every 4 units it moves across.
The amount per person is 300 ÷ 4 = 75 g, so for 10 people the recipe needs 75 × 10 = 750 g. This is a direct proportion, so scaling both quantities by the same factor keeps the ratio the same.
Subtract 7 from both sides to get 2x < 8, then divide both sides by the positive number 2 to get x < 4. Dividing by a positive number does not reverse the inequality sign.
Reflecting in the x-axis changes the sign of the y-coordinate while the x-coordinate stays the same. So (3, −2) becomes (3, 2), which is directly above the original point.
The volume of a cylinder is πr²h, so here it is 3.14 × 3² × 10 = 3.14 × 9 × 10 = 282.6 cm³. The radius is squared before multiplying by the height.
Multiply the coefficients: 2 × 4 = 8. Add the indices for each base: x³ × x² = x⁵ and y² × y⁵ = y⁷, giving 8x⁵y⁷.
The speed is 150 ÷ 2 = 75 km per hour, so in 5 hours the car covers 75 × 5 = 375 km. Distance is speed multiplied by time for constant speed.
The five numbers must total 5 × 12 = 60. The four known numbers add to 9 + 10 + 14 + 15 = 48, so the fifth number is 60 − 48 = 12.
This is a difference of two squares, since x² − 9 = x² − 3². It factorises as (x − 3)(x + 3), and expanding confirms the middle terms cancel to give x² − 9.
The cosine rule gives cos C = (a² + b² − c²) ÷ 2ab = (49 + 64 − 144) ÷ 112 = −31 ÷ 112 ≈ −0.277. Since the cosine is negative, the angle is obtuse, and cos⁻¹(−0.277) ≈ 113°.
From the second equation x = y + 2. Substituting into the first gives 3(y + 2) + 2y = 16, so 5y + 6 = 16 and y = 2, giving x = 4. Checking: 3(4) + 2(2) = 16 and 4 − 2 = 2.
The terms are 3, 6, 12, 24, 48 and 96, which add to 189. The formula Sₙ = a(rⁿ − 1)/(r − 1) gives 3(2⁶ − 1)/(2 − 1) = 3 × 63 = 189.
The x-coordinate of the vertex is −b/2a = 4/2 = 2. Substituting x = 2 gives y = 4 − 8 + 7 = 3, so the vertex is (2, 3), which is the minimum point since the parabola opens upwards.
Total surface area is πrl + πr² = 3.14 × 6 × 10 + 3.14 × 36 = 188.4 + 113.04 = 301.44 cm². The curved part uses the slant height, while the base uses the radius squared.
The fraction correct is 15/20, which simplifies to 3/4. Multiplying by 100 gives 75%, so the student answered 75% of the questions correctly.
A 15% discount removes 0.15 × 80 = $12 from the price, so the sale price is 80 − 12 = $68. Equivalently, the buyer pays 85% of the original price: 0.85 × 80 = $68.
Evaluate the powers first: 5² = 25 and 2³ = 8. Then subtract to get 25 − 8 = 17, following the order of operations where indices are worked out before subtraction.
Pythagoras' theorem gives a² + 5² = 13², so a² = 169 − 25 = 144 and a = 12 cm. The hypotenuse is always the longest side, so the missing side must be shorter than 13 cm.
Ordering the six values gives 3, 5, 7, 8, 9, 12. With an even number of values the median is the mean of the two middle values, (7 + 8) ÷ 2 = 7.5.
Rearranging into the form y = mx + c gives 2y = −4x + 10, so y = −2x + 5. The coefficient of x is the gradient, which is −2.
The outcomes greater than 4 are 5 and 6, giving 2 favourable outcomes out of 6 equally likely outcomes. So the probability is 2/6, which simplifies to 1/3.
Collect like terms: 5a + 2a = 7a and −3b + 7b = 4b. The simplified expression is therefore 7a + 4b, since unlike terms cannot be combined further.
Standard form writes a number as a value between 1 and 10 multiplied by a power of 10. Moving the decimal point four places to the right gives 4.6, so the power of ten is −4.
Using y = mx + c with m = 2 and the point (4, −3): −3 = 2(4) + c, so c = −11. The equation is y = 2x − 11, which can be checked by substituting x = 4 to give −3.
We need two numbers that multiply to give the constant 5 and add to give the coefficient of x, which is 6. The numbers 1 and 5 satisfy both conditions, giving (x + 1)(x + 5).
Arc length is the fraction of the circumference: (40/360) × 2 × π × 9 = (1/9) × 56.52 ≈ 6.28 cm. The angle fraction is one ninth of a full turn.
From 9 a.m. to 1 p.m. is 4 hours, so the population doubles 4 times. This gives 500 × 2⁴ = 500 × 16 = 8000 bacteria.
Of the 40 students, 24 walk and 40 − 24 = 16 take the bus. The ratio 24 : 16 simplifies by dividing both parts by 8 to give 3 : 2.
A fractional index means take a root and then a power: 27^(2/3) = (27^(1/3))². Since the cube root of 27 is 3, squaring gives 9.
If the car loses 20%, it retains 80% of its original value, so 0.8 × value = 18 000. Dividing gives 18 000 ÷ 0.8 = $22 500, which can be checked since 20% of 22 500 is 4500 and 22 500 − 4500 = 18 000.
The probability the first counter is red is 4/10 = 2/5. Given that, 3 red remain out of 9, so the second probability is 3/9 = 1/3. Multiplying gives (2/5) × (1/3) = 2/15.
The angles total 180°, and the ratio has 2 + 3 + 4 = 9 parts. Each part is 180° ÷ 9 = 20°, so the largest angle is 4 × 20° = 80°.
When multiplying powers with the same base, add the indices: x⁵ × x³ = x⁸. When dividing, subtract the indices: x⁸ ÷ x² = x⁶.
Let the width be w, so the length is w + 3 and w(w + 3) = 40. This gives w² + 3w − 40 = 0, which factorises as (w + 8)(w − 5) = 0, so w = 5 m since a width must be positive.
The slant height is the hypotenuse of a right triangle with legs equal to the radius and the height. Using Pythagoras: slant² = 5² + 12² = 25 + 144 = 169, so the slant height is 13 cm.
Multiply both sides by 4 to get 4y = 3x − 5, then add 5 to get 4y + 5 = 3x. Finally divide by 3, giving x = (4y + 5)/3.
The x-coordinate of the vertex is at x = −b/(2a) = 6/2 = 3. Substituting x = 3 gives y = 9 − 18 + 11 = 2, so the minimum point is (3, 2).
The midpoint is found by averaging the coordinates: x = (−2 + 4)/2 = 1 and y = (1 + 9)/2 = 5. So the midpoint is (1, 5).
The area is given by ½ab sin C, so ½ × 9 × 14 × sin 50° = 63 × 0.766 ≈ 48.26 cm². The sine formula is used when two sides and the included angle are known.
The terms are 5, −10, 20, −40 and 80. Adding them gives 5 − 10 + 20 − 40 + 80 = 55. The alternating signs partly cancel, leaving a positive total.