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In a base-ten number each place is ten times the place to its right, so the digit 7 sits in the hundreds place. Its value is therefore 7 × 100 = 700. The digit 4 in the thousands place is worth 4,000, while the 8 and 2 are worth 80 and 2.
List the factors of each number: 24 has 1, 2, 3, 4, 6, 8, 12, 24 and 36 has 1, 2, 3, 4, 6, 9, 12, 18, 36. The largest factor appearing in both lists is 12. Checking, 24 ÷ 12 = 2 and 36 ÷ 12 = 3, and no larger number divides both.
A fraction is a division, so 3/4 means 3 ÷ 4. Dividing gives 0.75 exactly. This is reasonable because 3/4 is close to 1 but less than 1, and 0.75 lies between 0.5 and 1.
Compare decimals place by place from left to right after lining up the decimal points. The thousandths digit decides the smallest value: 0.006 < 0.06 < 0.6 < 0.66. Writing them with equal length, 0.006, 0.060, 0.600, 0.660, makes the order clear.
An exponent tells how many times the base is used as a factor, so 2^5 = 2 × 2 × 2 × 2 × 2. Multiplying step by step gives 4, then 8, then 16, then 32. The result is 32, not 2 × 5.
A prime number has exactly two factors, 1 and itself. The number 29 is not divisible by 2, 3, or 5, and 5 × 5 = 25 < 29 < 36 = 6 × 6, so no factor below its square root divides it. Therefore 29 is prime.
Half of the recipe means multiplying the amount by 1/2, so compute 1/2 × 2/3. Multiply numerators and denominators: (1 × 2)/(2 × 3) = 2/6, which simplifies to 1/3. So half the recipe needs 1/3 cup.
List multiples of each number: 6, 12, 18, 24, 30 and 8, 16, 24, 32. The first multiple the two lists share is 24. Equivalently, LCM = (6 × 8) ÷ GCF = 48 ÷ 2 = 24.
Order of operations requires multiplication before addition and subtraction. First compute 3 × 2 = 6, giving 12 − 6 + 4. Working left to right, 12 − 6 = 6 and 6 + 4 = 10.
The discount is 25% of 40, which is 0.25 × 40 = 10 dollars. Subtract the discount from the original price: 40 − 10 = 30 dollars. A quick check is that the buyer pays 75% of 40, and 0.75 × 40 = 30.
The decimal 0.4 means 4 tenths, so it equals 4/10. Dividing numerator and denominator by 2 gives 2/5. Checking, 2 ÷ 5 = 0.4, so the fractions are equivalent.
Absolute value gives the distance from zero, so it is never negative. Thus |−7| = 7 and |3| = 3. Adding gives 7 + 3 = 10.
A number that is a multiple of both 4 and 6 must be a multiple of their least common multiple. Since LCM(4, 6) = 12, the multiples are 12, 24, 36, and so on. Among the choices, only 24 divides evenly by both 4 and 6.
The divisibility rule for 9 states that a number is divisible by 9 if and only if the sum of its digits is divisible by 9. For example, 738 has digit sum 18, and 738 ÷ 9 = 82. This works because 10, 100, and higher powers of ten all leave remainder 1 when divided by 9.
Rising means adding a positive change to the current depth, so compute −45 + 28. The result is −17 because the negative value has the larger magnitude: 45 − 28 = 17, and the sign stays negative. The submarine is still below sea level.
Each addend fills one place value: 5,000 supplies the thousands digit, 300 the hundreds digit, and 7 the ones digit. The tens place receives nothing, so it is filled with 0. The expanded form therefore equals 5,307.
Multiply the digits as whole numbers: 6 × 4 = 24. Each factor has one decimal place, so the product needs two decimal places. Placing the decimal point two places from the right gives 0.24.
Distribute the 3 first: 3(2x) + 3(4) = 6x + 12. Then combine like terms with −5x, giving 6x − 5x + 12 = x + 12. Only terms with the same variable part can be combined.
The total number of marbles is 3 + 5 + 4 = 12. There are 5 blue marbles, so the fraction is 5 out of 12, written 5/12. This fraction cannot be simplified because 5 and 12 share no common factor.
Let x = 0.454545…, then 100x = 45.454545…. Subtracting gives 99x = 45, so x = 45/99. Dividing numerator and denominator by 9 simplifies this to 5/11, which is the fraction in lowest terms.
Work backwards through the operations, reversing each one. Add 7 to 29 to get 36, then divide by 4 to get 9. Checking forward: 9 × 4 = 36 and 36 − 7 = 29, which matches.
Evaluate each expression: 2^3 × 3 = 8 × 3 = 24; 2 × 3^2 = 2 × 9 = 18; 3^3 = 27; 2^4 + 3 = 16 + 3 = 19. The largest of these values is 27, produced by 3^3.
The 10% increase gives 80 × 1.10 = 88 dollars. The 10% decrease applies to 88, not 80, so the discount is 8.80 dollars and the final price is 88 − 8.80 = 79.20 dollars. The two percentages are taken on different bases, so they do not cancel.
Use prime factorization: 84 = 2^2 × 3 × 7 and 126 = 2 × 3^2 × 7. The GCF takes the lowest power of each shared prime: 2 × 3 × 7 = 42. Checking, 84 ÷ 42 = 2 and 126 ÷ 42 = 3, so 42 divides both.
A fraction between 0 and 1 represents a part of a whole, so taking that part of a positive number yields a smaller result. For example, 12 × 1/2 = 6 and 12 × 3/4 = 9, both less than 12. Multiplying by exactly 1 keeps the number unchanged, and multiplying by more than 1 increases it.
A composite number has more than two factors. The number 21 equals 3 × 7, so it has factors 1, 3, 7, and 21, giving it more than two factors and making it composite.
The exponent tells how many times to use the base as a factor. Here 2⁴ means 2 × 2 × 2 × 2, which equals 16.
On a number line, values increase as you move to the right. Among negatives, −5 is farther left than −2, so −5 is the least, followed by −2, then the positive numbers 3 and 7.
Line up the decimal points and add place by place. Writing 4.5 as 4.50 gives 4.50 + 2.75 = 7.25.
The decimal 0.75 means 75 hundredths, or 75/100. Dividing the numerator and denominator by 25 gives 3/4, so 0.75 = 3/4.
Evaluate the parentheses first: 3 + 1 = 4. Then multiply 2 × 4 = 8, and finally subtract 15 − 8 = 7.
Moving to the right on a number line means adding. Starting at −3 and adding 5 gives −3 + 5 = 2.
To round to the nearest tenth, look at the hundredths digit. In 4.58 the hundredths digit is 8, so the tenths digit 5 rounds up to 6, giving 4.6.
Rewrite 1/4 as 2/8 so both fractions have the same denominator. Then 3/8 + 2/8 = 5/8.
The counts must be a common multiple of 6 and 8, and the smallest one is the least common multiple. Multiples of 8 are 8, 16, 24, and 24 is the first that is also a multiple of 6.
The product of two negative numbers is positive. Multiplying the absolute values gives 6 × 4 = 24, and the sign is positive.
Work backward from the result. Subtract 5 from 12 to get 7, then multiply by 3 to undo the division, giving 21. Checking: 21 ÷ 3 + 5 = 7 + 5 = 12.
The distributive property lets you split 46 into 40 + 6. Then 7 × 46 = 7 × (40 + 6) = 7 × 40 + 7 × 6.
Dividing by 0.2 is the same as multiplying by 5, because 0.2 = 1/5. So 0.8 ÷ 0.2 = 0.8 × 5 = 4.
All four numbers are negative, so the one closest to zero is greatest. Converting to decimals gives −0.5, −0.75, −0.4, and −0.9; the greatest is −0.4. Wait - −0.4 is greater than −0.5, so the correct answer is −0.4.
Convert 2 1/2 to the improper fraction 5/2, then multiply: 3/4 × 5/2 = 15/8. This equals 1 7/8 cups.
Multiplication comes before subtraction. First 3 × (−2) = −6, then 5 − (−6) = 5 + 6 = 11.
Multiply the numerator and denominator by 5 to make the denominator 100: 7/20 = 35/100 = 0.35.
Because 100 is divisible by 4, only the last two digits determine divisibility by 4. For example, 1,316 ends in 16, and 16 is divisible by 4, so 1,316 is divisible by 4.
When multiplying powers with the same base, add the exponents. So 2³ × 2² = 2^(3+2) = 2⁵, which equals 32.
Adding the deposit gives −35 + 50 = 15. Then subtracting the fee gives 15 − 12 = 3, so the balance is 3 dollars.
The prime factorizations are 48 = 2⁴ × 3 and 72 = 2³ × 3². Taking the lowest power of each shared prime gives 2³ × 3 = 24.
Increasing 60 by 20% gives 60 × 1.2 = 72. Decreasing 72 by 20% gives 72 × 0.8 = 57.60, which is less than the original price.
Dividing by a fraction means multiplying by its reciprocal. So (3/5) ÷ (9/10) = (3/5) × (10/9) = 30/45 = 2/3.
Let x = 0.272727… Then 100x = 27.272727…, and subtracting gives 99x = 27, so x = 27/99 = 3/11.