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Reading place value from the right, the digits name ones, tens, hundreds, thousands, and ten-thousands. In 48,372 the thousands digit is 8, while 4 sits in the ten-thousands place.
In a decimal, the first place after the point is tenths, the second is hundredths, and the third is thousandths. The 6 in 5.064 stands in the third decimal place, so its value is 6 thousandths, or 0.006.
Compare decimals place by place from the left. All four have 0 in the ones place and 6 in the tenths place, so compare hundredths: 0.66 has 6 hundredths while the others have 0, 0, and 0, making 0.66 the largest.
Rewrite each value as a decimal: 2/3 is about 0.667, 0.7 is 0.700, and 3/4 is 0.75. Comparing the decimal forms gives 0.667 < 0.700 < 0.750, so the order is 2/3, 0.7, 3/4.
A prime number has exactly two factors, 1 and itself. The number 29 is not divisible by 2, 3, or 5, and no factor between 2 and 5 divides it, so it is prime.
List the factors of each number: 18 has 1, 2, 3, 6, 9, 18 and 24 has 1, 2, 3, 4, 6, 8, 12, 24. The largest factor appearing in both lists is 6.
The decimal 0.25 means 25 hundredths, which simplifies to 1/4 because 25 divided by 25 is 1 and 100 divided by 25 is 4. Checking by division, 1 divided by 4 equals 0.25.
Each addend gives the digit for one place value: 4 thousands, 3 hundreds, 2 tens, and 7 ones. Writing those digits in order from left to right gives 4,327.
Rounding to the nearest hundred gives 500 for numbers from 450 through 549, and rounding to the nearest ten gives 460 for numbers from 455 through 464. The only value in both ranges is 462.
Rewrite 1/4 as 2/8 so both fractions have the same denominator, then add the numerators: 3/8 + 2/8 = 5/8. The denominator stays the same when adding fractions with like denominators.
Convert 1/3 to sixths by multiplying numerator and denominator by 2, giving 2/6. Then 5/6 - 2/6 = 3/6, which simplifies to 1/2.
Multiply numerators and denominators: 2 x 3 = 6 and 3 x 4 = 12, giving 6/12. Simplifying by dividing both parts by 6 gives 1/2.
Dividing by a fraction is the same as multiplying by its reciprocal, so 3/5 divided by 1/10 equals 3/5 x 10/1. That product is 30/5, which simplifies to 6.
Multiply as with whole numbers: 6 x 4 = 24. Since the factors have one decimal place each, the product needs two decimal places, giving 0.24.
Multiply both numbers by 10 to remove the decimal from the divisor: 72 divided by 8. That quotient is 9, and scaling both numbers by the same factor keeps the quotient unchanged.
Adding a zero to the right of the last decimal digit does not change the value, so 3.6 and 3.60 name the same number. Both have 3 ones and 6 tenths.
Multiply the amount per batch by the number of batches: 3 x 2 1/2. That is 3 x 2 = 6 plus 3 x 1/2 = 1 1/2, giving 7 1/2 cups.
Since 6 equals 2 times 3, any number divisible by 6 is divisible by both 2 and 3. So divisibility by 3 is guaranteed, while the other statements are not.
A common multiple must be divisible by both numbers. The number 24 divides evenly by 4 (6 times) and by 6 (4 times), so it is a common multiple.
Divide the numerator by the denominator: 7 divided by 8. Since 1/8 equals 0.125, seven eighths equals 7 x 0.125 = 0.875.
Build the number place by place: 9 ones gives 9, 0 tenths gives .0, 4 hundredths gives .04, and 6 thousandths gives .006. Combining them in order gives 9.046.
A number with exactly three factors must be the square of a prime, because its factors are 1, the prime, and the square. Since 9 equals 3 x 3, its factors are 1, 3, and 9.
List multiples of 12 (12, 24, 36, 48, ...) and of 18 (18, 36, 54, ...). The smallest number appearing in both lists is 36, which is the least common multiple.
Convert each fraction to a decimal: 7/8 equals 0.875, which is greater than 0.85. Checking the others shows 3/5 = 0.6, 2/3 is about 0.667, and 5/8 = 0.625.
Rounding to the nearest tenth gives 3.7 for numbers from 3.65 through 3.74, and rounding to the nearest whole number gives 4 for numbers from 3.5 through 4.49. The value 3.72 lies in both ranges.
Expanded form lists the value of each place, so the digits are 5 in the ten-thousands place, 6 in the thousands place, 4 in the hundreds place, 3 in the tens place, and 2 in the ones place. Putting these digits in order gives 56,432.
A prime number has exactly two factors: 1 and itself. The number 29 has no whole-number factors other than 1 and 29, so it is prime.
In 4,738,215 the digit 7 sits in the hundred-thousands place, counting from the right: ones, tens, hundreds, thousands, ten-thousands, hundred-thousands. Its value is therefore 7 times 100,000, which is 700,000.
A factor divides a number with no remainder. Nine divides 36 evenly because 9 times 4 equals 36, but 24 divided by 9 leaves a remainder of 6.
The divisibility rule for 9 states that a number is divisible by 9 exactly when the sum of its digits is a multiple of 9. For example, 738 has digit sum 18, and 738 divided by 9 equals 82.
A fraction is greater than 1 when its numerator is larger than its denominator. In 9/4 the numerator 9 exceeds the denominator 4, so the fraction equals 2 1/4, which is greater than 1.
Dividing a whole number by 100 shifts every digit two places to the right. The 6 in the thousands place moves to the tens place, giving 60, which checks out because 60 times 100 equals 6,000.
List the factors the two numbers share: 1, 2, 3, 4, 6, and 12. The largest of these common factors is 12, and it checks because 36 equals 12 times 3 and 60 equals 12 times 5.
Multiples of 8 are 8, 16, 24, 32, 40, and multiples of 10 are 10, 20, 30, 40. The first number appearing in both lists is 40, so the least common multiple is 40.
Rounding to the nearest tenth means keeping one decimal place and checking the hundredths digit. The number 8.46 has a hundredths digit of 6, so the tenths digit rounds up from 4 to 5, giving 8.5.
Rewrite 2/5 as 4/10 so both fractions share the denominator 10. Then add the numerators: 4/10 plus 3/10 equals 7/10, which is already in simplest form.
Use the least common denominator, 12: 5/6 becomes 10/12 and 1/4 becomes 3/12. Subtracting gives 7/12, which cannot be simplified further because 7 and 12 share no factor other than 1.
Multiplying a fraction by a whole number multiplies the numerator: 3 times 8 equals 24, so the product is 24/4. Dividing 24 by 4 gives 6.
To divide by a fraction, multiply by its reciprocal. So 4/5 times 3/2 equals 12/10, which simplifies to 6/5, or 1 1/5.
The decimal 0.35 means 35 hundredths, or 35/100. Dividing the numerator and denominator by their greatest common factor, 5, gives 7/20.
Multiply 7 by 3 to get 21, then count the decimal places in the factors: one in 0.7 and two in 0.03, for three total. Placing the decimal point three places from the right in 21 gives 0.021.
Multiply both numbers by 10 to remove the decimal from the divisor: 45 divided by 9. That quotient is 5, and scaling both numbers by the same power of ten leaves the quotient unchanged.
A number is divisible by 3 when its digit sum is a multiple of 3, and by 9 when its digit sum is a multiple of 9. The digits of 123 sum to 6, which is a multiple of 3 but not of 9.
Rounding to the nearest hundred gives 700 for numbers from 650 to 749, and rounding to the nearest ten gives 650 for numbers from 645 to 654. The number 649 lies in both ranges, so it satisfies both conditions.
Divide the numerator by the denominator: 5 divided by 8. Since 8 times 0.6 equals 4.8 and 8 times 0.025 equals 0.2, the sum is 0.625.
The factors of 18 are 1, 2, 3, 6, 9, and 18, which is six factors, so 18 does not work; the factors of 12 are 1, 2, 3, 4, 6, and 12, also six. Checking 24 gives eight factors, but 18 is not the answer. The number that has exactly four factors and includes 6 is 12 only when paired with another prime, so the correct choice is 18.
The numbers 3 and 8 share no factor other than 1, so their greatest common factor is 1. Their least common multiple is 24 because 24 is the smallest number divisible by both 3 and 8.
Work backward through the operations. The final value 0.45 came from dividing by 100, so multiply by 100 to get 45; that 45 came from multiplying by 10, so divide by 10 to get 4.5.
Cross-multiply to compare: 3 times 9 equals 27 and 4 times 7 equals 28, so 4/9 is actually larger. Comparing to one half shows the reasoning: 3/7 is just under 1/2 while 4/9 is just under 1/2 as well, and 4/9 is closer. The comparison 27 against 28 shows that 4/9 exceeds 3/7, so neither the size of the numerators nor the size of the denominators decides alone.
Rounding to the nearest thousand gives 5,000 for numbers from 4,500 through 5,499. The number 4,501 lies in that range because its hundreds digit is 5, which rounds the thousands digit up.