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A ratio of boys to girls of 3:5 means the fraction of boys over girls equals 3/5. Setting the known count 12 boys over the unknown number of girls g gives 3/5 = 12/g, which solves to g = 20. The parts must stay in the same order on both sides of the proportion.
'Three times a number n' is 3n, and 'seven less than' that quantity means subtract 7 from it, giving 3n - 7. The order matters because subtraction is not commutative. Reading 'less than' as reversing the subtraction is the key step.
When multiplying powers with the same base, add the exponents: 2^3 × 2^4 = 2^(3+4) = 2^7. The base stays the same; only the exponents combine. This rule applies because both factors share base 2.
Slope equals the change in y divided by the change in x: (11 - 3)/(6 - 2) = 8/4 = 2. Always subtract coordinates in the same order to keep signs consistent. The result 2 means y increases 2 units for every 1 unit increase in x.
An irrational number cannot be written as a ratio of two integers and has a non-terminating, non-repeating decimal. The square root of 2 is the classic example, since no fraction squared equals exactly 2. The other choices are all rational.
A 25% discount removes 0.25 × 40 = $10 from the original price, leaving 40 - 10 = $30. Equivalently, the customer pays 75% of $40, which is also $30. The discount amount is not the final price.
With an even number of values, the median is the mean of the two middle values. Here the middle values are 9 and 12, so the median is (9 + 12)/2 = 10.5. The data are already in order, so no sorting is needed.
Combine like terms separately: 4x - 2x = 2x and 3y + 5y = 8y, giving 2x + 8y. Terms with different variables cannot be merged into a single term. The order of the resulting terms does not affect the value.
By the Pythagorean theorem, the hypotenuse c satisfies c² = 6² + 8² = 36 + 64 = 100, so c = 10. This is the familiar 3-4-5 triangle scaled by 2. The hypotenuse is always the longest side but is not the sum of the legs.
Distribute to get 3x - 12 = 2x + 5. Subtract 2x from both sides to get x - 12 = 5, then add 12 to obtain x = 17. Checking: 3(13) = 39 and 2(17) + 5 = 39, so the solution is consistent.
There are 7 blue marbles out of a total of 5 + 7 = 12 marbles, so the probability of blue is 7/12. Probability compares favorable outcomes to all equally likely outcomes. The denominator must be the total count, not one color's count.
The y-intercept occurs where x = 0, so substitute to get f(0) = 2(0) + 1 = 1, giving the point (0, 1). In slope-intercept form y = mx + b, the constant b is the y-intercept. Here b = 1.
We need two numbers that multiply to 6 and add to -5, namely -2 and -3. Therefore x² - 5x + 6 = (x - 2)(x - 3). Checking by expanding gives x² - 5x + 6, confirming the factorization.
Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of -3/4 is 4/3, since (-3/4)(4/3) = -1. The product of perpendicular slopes must equal -1.
The average rate is 150/3 = 50 miles per hour. In 5 hours the car covers 50 × 5 = 250 miles. Setting up a proportion such as 150/3 = d/5 yields the same result.
When dividing powers with the same base, subtract the exponents: x^5 / x^2 = x^(5-2) = x^3. The base stays the same, and only the exponents are combined by subtraction. This requires x ≠ 0 to avoid division by zero.
The total of all five scores must be 5 × 82 = 410. The four known scores sum to 78 + 85 + 90 + 76 = 329. The fifth score is 410 - 329 = 81.
Area equals length times width, so multiply (x + 3)(x - 2) using FOIL: x² - 2x + 3x - 6 = x² + x - 6. Combining the middle terms gives the positive x coefficient. The constant term is the product of the constants, -6.
Adding the two equations eliminates y: 3x = 9, so x = 3. Substituting into x - y = 2 gives 3 - y = 2, so y = 1. Checking in 2x + y = 7: 6 + 1 = 7, which is consistent.
The angles of a triangle sum to 180 degrees, so 2x + 3x + 4x = 9x = 180. Dividing gives x = 20. The angles are then 40, 60, and 80 degrees, which sum to 180.
Each term is doubled, so the common ratio is 2. The 5th term is 48 and the 6th term is 48 × 2 = 96. Using the formula a·r^(n-1) gives 3 × 2^5 = 96.
Subtract 5 from both sides to get -2x > 6. Dividing both sides by -2 reverses the inequality sign, giving x < -3. Reversing the sign is required whenever multiplying or dividing by a negative number.
The numerator factors as a difference of squares: x² - 9 = (x - 3)(x + 3). Canceling the common factor (x - 3) leaves x + 3, valid for x ≠ 3. The original expression still has a removable discontinuity at x = 3.
The volume of a cone is V = (1/3)πr²h. Substituting r = 3 and h = 7 gives (1/3)π(3²)(7), which simplifies to 21π cubic centimeters. The one-third factor distinguishes a cone from a cylinder.
The vertex x-coordinate is -b/(2a) = 6/2 = 3. Factoring x² - 6x + 5 gives (x - 1)(x - 5), so the roots are x = 1 and x = 5. The vertex lies midway between the roots, consistent with x = 3.
A 15% markup multiplies the price by 1.15, giving $69.00. The 20% discount then multiplies that amount by 0.80, so 69 × 0.80 = 55.20. Successive percent changes multiply rather than add, which is why the answer is not $57.00 from a net 5% decrease.
The original nine values total 9 × 12 = 108. Adding 32 gives a new total of 140 across 10 values, so the new mean is 140 ÷ 10 = 14. Recomputing the total from the original mean is the key step.
Distributing gives 10x - 15 - 4x + 24, because subtracting -24 becomes adding 24. Combining like terms yields 6x + 9. The sign change on the second distribution is the step most often missed.
The ladder, wall, and ground form a right triangle with the ladder as the hypotenuse. Using the Pythagorean theorem, the height is √(13² - 5²) = √(169 - 25) = √144 = 12 feet.
Substituting the first equation into the second gives 3x + (2x - 1) = 9, so 5x = 10 and x = 2. Substitution works cleanly here because the first equation already isolates y.
The interior angles of any quadrilateral sum to 360°. The three known angles total 290°, so x = 360 - 290 = 70.
Applying the rule step by step: a₂ = 3(4) - 2 = 10, a₃ = 3(10) - 2 = 28, and a₄ = 3(28) - 2 = 70. Each term depends on the immediately preceding term, so the recursion must be evaluated in order.
The first inequality gives x < 5, and the second gives x ≥ 2, so the solution set is 2 ≤ x < 5. Only 2 lies in that interval.
Factoring the numerator gives (2x - 3)(x + 5), since 2x² + 10x - 3x - 15 matches the original. Canceling (x + 5) leaves 2x - 3.
The volume of a cylinder is V = πr²h. Substituting r = 4 and h = 10 gives π(16)(10) = 160π cubic centimeters.
Completing the square gives f(x) = (x - 2)² + 3, so the vertex is (2, 3). Because the parabola opens upward, the vertex supplies the minimum value of 3.
There are 20 candies in total and 6 are yellow, so the probability is 6/20, which reduces to 3/10. The denominator must count every candy, not just the non-yellow ones.
The slope between the points is (13 - 5) ÷ (6 - 2) = 8 ÷ 4 = 2. Parallel lines share the same slope, so the answer is 2.
Depreciation by 12% leaves 88% of the value each year, so the multiplier is 0.88. Applying it for three consecutive years gives 20,000(0.88)³, which is exponential decay rather than repeated subtraction.
The ratio 3:5 means cats make up 3 parts for every 5 parts of dogs. With 40 dogs, each part is 40 ÷ 5 = 8 animals, so cats number 3 × 8 = 24.
Consecutive even integers differ by 2, so the three are 2n, 2n + 2, and 2n + 4. Adding them gives 6n + 6, which is always even as expected.
For a square with side s, the diagonal is s√2, so s = 10/√2. The area is s² = 100/2 = 50 square units.
Evaluating each power gives 9 + 8 - 1 = 16. Any nonzero number raised to the zero power equals 1, which is the step most often overlooked.
The absolute value |x - 3| is never negative, so its smallest value is 0, occurring at x = 3. Adding 2 shifts the minimum to 2, making the range all values greater than or equal to 2.
Solving for y: -2y = -4x + 10, so y = 2x - 5. The coefficient of x is the slope, 2, and the constant term is the y-intercept, -5.
The chance the first part works is 9/12 = 3/4. Given that, 8 of the remaining 11 work, so the probability is (3/4)(8/11) = 24/44 = 6/11. Without replacement, the second draw uses updated counts.
The distance formula gives √((4 - 1)² + (6 - 2)²) = √(9 + 16) = √25 = 5. Squaring the differences before adding is essential.
This is a difference of cubes with a = 2x and b = 3, using a³ - b³ = (a - b)(a² + ab + b²). Substituting gives (2x - 3)(4x² + 6x + 9).
Areas of similar figures scale as the square of the side ratio, so the area ratio is 4:25. Multiplying 12 by 25/4 gives 75 square units.
Walking and bus riders together make up 75% of the students, leaving 25% who are driven. Then 0.25 × 200 = 50 students.