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The digit 7 sits in the hundred-thousands place, so its value is 7 × 100,000 = 700,000. Place value is determined by counting positions outward from the decimal point, not by the digit's face value. The 4 in the millions place contributes 4,000,000, which is a different position entirely.
The hundreds digit is 8, and the digit to its right in the tens place is 4. Because 4 is less than 5, the hundreds digit stays the same and the tens and ones digits become zeros, giving 5,800. Rounding to a place keeps all digits to the left of that place unchanged unless the next digit is 5 or greater.
Order of operations requires multiplication before addition and subtraction, so compute 3 × 2 = 6 first. The expression becomes 12 − 6 + 4, and addition and subtraction are then performed left to right: 12 − 6 = 6, then 6 + 4 = 10. Working strictly left to right without honoring the multiplication is the common error.
Adding a positive number to a negative number moves rightward on the number line. Start at −7 and move 12 units right, which lands on 5. Equivalently, 12 − 7 = 5, and the result takes the sign of the addend with the larger absolute value, which is positive.
Line up the decimals and compare place by place. Writing the numbers as 0.650, 0.605, 0.600, and 0.060 shows that 0.650 has the largest hundredths digit among those with 6 in the tenths place, while 0.060 is smallest. Therefore 0.65 is the greatest of the four.
To convert a decimal to a percent, multiply by 100 and attach the percent symbol, which moves the decimal point two places to the right. So 0.6 × 100 = 60, giving 60%. Dividing by 100 instead would produce 0.006, a common reversal error.
Write 0.375 as 375/1000 and simplify by dividing numerator and denominator by 125, which gives 3/8. Checking by division confirms that 3 ÷ 8 = 0.375. A denominator of 8 fits a terminating decimal because 8 has only 2 as a prime factor.
Absolute value gives the distance of a number from zero on the number line, so it is never negative. The number −9 is 9 units from 0, so |−9| = 9. The negative sign is removed rather than squared away.
Multiples of 12 are 12, 24, 36, 48, ... and multiples of 18 are 18, 36, 54, ..., so the smallest shared multiple is 36. Using prime factorizations, 12 = 2² × 3 and 18 = 2 × 3², so the LCM is 2² × 3² = 36. The LCM is always at least as large as the greater of the two numbers.
The prime factorizations are 24 = 2³ × 3 and 36 = 2² × 3². The greatest common factor uses the lower power of each shared prime, giving 2² × 3 = 12. Every common factor of 24 and 36 divides 12 evenly.
Scientific notation writes a number as a × 10^n with 1 ≤ a < 10. Moving the decimal point in 0.00042 four places right produces 4.2, and because the point moved right the exponent is −4, giving 4.2 × 10^−4. Checking: 4.2 × 10^−4 = 0.00042.
Convert the percent to a decimal, 0.15, and multiply: 0.15 × 240 = 36. A mental check is that 10% of 240 is 24 and 5% is 12, and 24 + 12 = 36. The answer must be less than 240 because the percent is below 100.
Set up the proportion 3/12 = x/30 because the ratio of cups to cookies stays constant. Cross-multiplying gives 12x = 90, so x = 7.5. Equivalently, 30 cookies is 2.5 times 12 cookies, and 2.5 × 3 = 7.5.
Percent change equals the amount of change divided by the original amount, times 100. The change is 50 − 40 = 10, and 10 ÷ 40 = 0.25, which is a 25% increase. Dividing by the new amount instead of the original is the common mistake that produces 20%.
Subtracting a negative number is the same as adding its opposite, so −5 − (−8) = −5 + 8. Starting at −5 and moving 8 units right gives 3. Ignoring the double negative and adding absolute values would give the wrong sign.
When multiplying powers with the same base, add the exponents: 2^5 × 2^3 = 2^(5+3) = 2^8, which equals 256. Multiplying the exponents is the rule for a power raised to a power, not for a product of powers.
Since 7² = 49 and 8² = 64, and 49 < 50 < 64, the square root of 50 lies between 7 and 8. It is only slightly greater than 7 because 50 is just above 49. The value is irrational, so it is not a whole number.
The distributive property multiplies the factor outside the parentheses by each term inside: 6 × 2x = 12x and 6 × 5 = 30. The simplified expression is therefore 12x + 30. Multiplying only the first term is a frequent error.
A unit rate compares the cost to one pound, so divide $12.50 by 5 pounds: 12.50 ÷ 5 = 2.50. The unit price is $2.50 per pound. Multiplying instead of dividing would just return the cost of five pounds.
Dividing by a fraction is the same as multiplying by its reciprocal, so 2/3 ÷ 4/9 = 2/3 × 9/4. Multiplying numerators and denominators gives 18/12, which simplifies to 3/2. The quotient is greater than 1, as expected because 2/3 is larger than 4/9.
Let x = 0.444... Then 10x = 4.444..., and subtracting gives 9x = 4, so x = 4/9. The repeating block has one digit, which is why multiplying by 10 isolates it. Checking, 4 ÷ 9 = 0.444....
A rational number can be written as a ratio of two integers: 0.75 = 3/4, −3 = −3/1, and 22/7 is already such a ratio. The square root of 2 cannot be written that way because 2 is not a perfect square, so its decimal neither terminates nor repeats.
A negative exponent indicates the reciprocal of the base raised to the positive exponent: (2/3)^−2 = (3/2)². Squaring gives 9/4. The result is positive because the exponent is even and the base is positive.
After a 20% discount, the sale price is 80% of the original, so 0.80x = 48. Dividing both sides by 0.80 gives x = 60. Adding 20% of the sale price instead of 20% of the original is the common error that produces $57.60.
Divide the coefficients and subtract the exponents: 6 ÷ 2 = 3 and 10^7 ÷ 10^3 = 10^4. The result is 3 × 10^4, or 30,000. The quotient should be far smaller than 6 × 10^7, which confirms that the exponents were subtracted.
On a horizontal number line, values increase from left to right, and −6 lies to the right of −8, so −6 is greater than −8. Absolute value measures distance from zero, and the distance for −8 is larger than the distance for −6.
Reading the place values from the right, the digits are 5 in the ones place, 1 in the tens, 9 in the hundreds, 8 in the thousands, 2 in the ten-thousands, 7 in the hundred-thousands, and 4 in the millions. The digit 7 therefore names a count of hundred-thousands.
Rewrite both fractions with the least common denominator 12: 1/4 = 3/12 and 5/6 = 10/12. Adding the numerators gives 13/12, which equals 1 1/12.
Multiplying the first two factors gives (−4)(−6) = 24, because the product of two negatives is positive. Then 24 × (−2) = −48, because a positive times a negative is negative.
When multiplying powers with the same base, add the exponents, and when dividing, subtract them: 7 + (−3) − 2 = 2. The expression therefore simplifies to 5^2, which equals 25.
Set up the proportion 18/45 = p/100, since a percent compares a part with 100. Dividing gives 18 ÷ 45 = 0.4, which is 40%.
Set up the proportion 1 inch / 25 miles = 3.5 inches / d miles and multiply across: d = 3.5 × 25 = 87.5. The road is 87.5 miles long.
Multiply as if the numbers were whole numbers: 6 × 45 = 270. The factors have one and two decimal places, three in total, so place the decimal point three places from the right to get 0.270 = 0.27.
The next simultaneous arrival happens after the least common multiple of 9 and 15 minutes. Since 9 = 3² and 15 = 3 × 5, the LCM is 3² × 5 = 45 minutes, which gives 8:45 a.m.
Distance is the absolute value of the difference of the coordinates: |4 − (−7)| = |11| = 11. Computing |−7 − 4| gives the same result, since distance is never negative.
Factor 84 as 4 × 21, then write each factor in prime form: 4 = 2² and 21 = 3 × 7. The prime factorization is 2² · 3 · 7, which checks because 4 × 3 × 7 = 84.
Following the order of operations, evaluate the exponent first: 4² = 16. Then multiply: 2 × 16 = 32, and finally add: 3 + 32 = 35.
Subtract inside the parentheses using the common denominator 10: 2/5 = 4/10 and 1/2 = 5/10, so the difference is −1/10. Dividing by 3/10 means multiplying by its reciprocal 10/3, so (−1/10) × (10/3) = −1/3.
Represent each change with a multiplier: a 25% increase gives 1.25, and a 20% decrease gives 0.8. Since 1.25 × 0.8 = 1.00, the final price equals the original price, so there is no net change.
To add numbers in scientific notation, first express both with the same power of ten: 3.0 × 10^4 = 0.30 × 10^5. Then add the coefficients: (4.2 + 0.30) × 10^5 = 4.5 × 10^5.
Order of operations requires exponents first, then multiplication and division from left to right, then addition and subtraction. Here 2² = 4, so the expression becomes 3 + 16 − 2 = 17.
Combine like terms by adding the coefficients of terms with the same variable: (5x − 2x) + (3y + 7y) = 3x + 10y. Terms with different variables cannot be merged into one term.
"Three times a number" is 3n, and "seven less than" that quantity means subtracting 7 from 3n. The correct translation is therefore 3n − 7.
Add 9 to both sides to isolate the variable term, giving 4x = 32. Dividing both sides by 4 gives x = 8.
In Quadrant II, x-coordinates are negative and y-coordinates are positive. The point (−5, 3) has a negative x-value and a positive y-value, so it lies in Quadrant II.
When dividing powers with the same base, subtract the exponents: x⁹ ÷ x³ = x^(9−3) = x⁶.
Distribute −3 to each term inside the parentheses: −3(2x) = −6x and −3(−5) = +15. The expanded form is −6x + 15.
The x-intercept occurs where y = 0. Substituting y = 0 into 2x + 3y = 12 gives 2x = 12, so x = 6 and the intercept is (6, 0).
Slope equals the change in y divided by the change in x: (−7 − 5)/(4 − (−2)) = −12/6 = −2.
Using point-slope form, y − 5 = −3(x − 2). Distributing gives y − 5 = −3x + 6, so y = −3x + 11.