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In 4,706 the digit 7 sits in the hundreds place, so its value is 7 hundreds, or 700. Place value is determined by the position of the digit relative to the decimal point.
'Three hundred forty thousand' gives 340,000, and 'twenty-six' fills the ones and tens places with 26, while the hundreds place of the thousands group is zero. Writing the groups together gives 340,026.
An irrational number cannot be written as a ratio of two integers, and its decimal form neither terminates nor repeats. The square root of 2 is a classic example of such a number.
Multiplication is performed before addition, so 3 x 4 = 12 first. Then 5 + 12 = 17.
The decimal 0.6 means 6 tenths, which is 6/10. Reducing 6/10 by dividing numerator and denominator by 2 gives 3/5 in simplest form.
On a number line, values increase as you move right, and negative numbers with larger absolute values are farther left. So -7 is least, followed by -3, then 0, then 4.
The factors common to both numbers are 1, 2, 3, 4, 6, and 12, and the largest of these is 12. Another route is to factor 24 = 2^3 x 3 and 36 = 2^2 x 3^2, then take 2^2 x 3 = 12.
Multiplying 2/3 by 4 means multiplying the numerator by 4, giving 8/3. This improper fraction equals 2 2/3 cups.
Rewrite 1/4 as 2/8 so the denominators match, then subtract the numerators: 7/8 - 2/8 = 5/8. The result is already in simplest form.
Multiplying by 10^3 moves the decimal point three places to the right. Starting at 3.6, that yields 3,600.
The unit price is $4.50 divided by 3, which is $1.50 per notebook. Multiplying $1.50 by 8 gives $12.00.
Twenty-five percent equals one quarter, so divide 240 by 4 to get 60. Equivalently, 0.25 x 240 = 60.
The discount is 0.30 x $80 = $24, and subtracting that from $80 gives $56. Alternatively, paying 70% of $80 also yields $56.
A prime number has exactly two distinct positive factors, 1 and itself. The number 97 is not divisible by 2, 3, 5, or 7, and no other prime up to its square root divides it, so it is prime.
Multiples of 6 include 6, 12, 18, 24, and multiples of 8 include 8, 16, 24, so the smallest shared multiple is 24. Using prime factors, 6 = 2 x 3 and 8 = 2^3, and 2^3 x 3 = 24.
Adding two negative numbers gives a negative sum whose absolute value is the sum of the absolute values. Since 5 + 8 = 13, the result is -13.
Moving left on a number line means subtracting, so -2 - 3 = -5. The point lies five units left of zero.
The distributive property states that a x (b + c) = a x b + a x c. Applying it, 4 x (6 + 3) equals 4 x 6 + 4 x 3, both of which equal 36.
Dividing 5 by 8 gives 0.625, a terminating decimal. The denominator 8 has only 2 as a prime factor, so the decimal terminates.
Comparing a decimal to a fraction requires a common form; 1/2 equals 0.5, and 0.45 is less than 0.5. The student compared digits and whole numbers without regard to place value or the value of the fraction.
When multiplying powers with the same base, add the exponents: 4 + 3 = 7, giving 2^7. This equals 128.
Converting to decimals, 3/5 = 0.6 and 7/10 = 0.7, so a value between them must lie strictly between 0.6 and 0.7. Only 0.65 satisfies this.
Since 6 = 2 x 3 and 2 and 3 are relatively prime, a number is divisible by 6 exactly when it is divisible by both 2 and 3. Testing divisibility by 2 and 3 together is therefore sufficient.
Increasing $40 by 25% gives $50. Decreasing $50 by 20% subtracts $10, giving $40, so the final price equals the original. The two percent changes act on different bases, which is why they do not cancel to zero change in general.
Since 7^2 = 49 and 50 is greater than 49, the square root of 50 is slightly greater than 7; however, 7.1^2 = 50.41, so the root lies between 7 and 7.1, meaning it is less than 7 only if the comparison is reversed. Careful: the correct ordering is that the root of 50 exceeds 7, so compare by squaring both sides: 50 > 49 implies root 50 > 7.
In a base-ten numeral, each position is ten times the value of the position to its right. In 4,782 the digits from left to right sit in the thousands, hundreds, tens, and ones places, so the 7 occupies the hundreds place. Its value is therefore 7 × 100 = 700.
To round to the nearest hundred, locate the hundreds digit (4) and check the digit to its right (5). A tens digit of 5 or more rounds the hundreds digit up, so 6,458 rounds to 6,500. The student looked only at the hundreds digit and ignored the deciding tens digit.
Compare place values from left to right after aligning the decimals. Both numbers have 0 in the tenths place? No: 0.6 has 6 tenths and 0.58 has 5 tenths, so 0.6 is larger. Writing 0.6 as 0.60 makes the comparison clear: 60 hundredths exceeds 58 hundredths.
A repeating decimal can be converted by setting x = 0.333... and multiplying by 10 to get 10x = 3.333.... Subtracting gives 9x = 3, so x = 3/9 = 1/3. Checking by division confirms that 1 ÷ 3 produces the repeating digit 3 indefinitely.
Dividing by a fraction is the same as multiplying by its reciprocal, so 5/6 ÷ 1/3 = 5/6 × 3/1. Multiplying numerators and denominators gives 15/6, which simplifies to 5/2. The result makes sense because 1/3 fits into 5/6 two and a half times.
The fraction 7/8 means 7 divided by 8, so the decimal d must satisfy 8 × d = 7. Multiplying 0.875 by 8 gives exactly 7, confirming the conversion. This inverse-operation check is a reliable way to verify any fraction-to-decimal conversion.
The least common denominator of 4 and 6 is 12, so rewrite 3/4 as 9/12 and 5/6 as 10/12. Adding the numerators gives 19/12, which is greater than 1 and can also be written as 1 7/12.
The number described is 27, since its factors are exactly 1, 3, 9, and 27. A number with a factor other than 1 and itself is composite, and 27 = 3 × 3 × 3, so it is a power of 3. It is not prime because it has factors besides 1 and 27.
Order of operations requires multiplication before addition, so 8 + 4 × 2 = 8 + 8 = 16. Grouping symbols around the multiplication, as in 8 + (4 × 2), preserve this standard order and also give 16. The other groupings change which operation is performed first.
The distributive property states that a × (b + c) = (a × b) + (a × c), so a factor multiplied by a sum equals the sum of the individual products. Here the factor 6 is distributed to both 5 and 9. The equality holds because multiplication is being spread across an addition.
Starting 12 meters below sea level is -12. Rising 5 meters gives -12 + 5 = -7, and descending 9 meters gives -7 - 9 = -16. The final position is 16 meters below sea level, written as the integer -16.
The product of two negative integers is positive, so (-4) × (-3) = 12. Multiplying by the third negative factor gives 12 × (-2) = -24. With three negative factors, the product is negative because an odd number of negatives yields a negative result.
The absolute value of -7 is 7, since absolute value gives distance from zero. The opposite of 7 is -7, found by changing its sign. Combining the two ideas, the opposite of |-7| is -7.
Multiplying by 10^4 moves the decimal point four places to the right. Working backward, divide 0.35 by 10,000, which moves the decimal point four places to the left, giving 0.000035. Checking forward, 0.000035 × 10,000 = 0.35.
Scientific notation writes a number as a factor between 1 and 10 multiplied by a power of 10. Moving the decimal point in 0.00062 four places to the right gives 6.2, so the exponent must be -4 to keep the value unchanged. Thus 0.00062 = 6.2 × 10^-4.
A 15% discount leaves 85% of the price, so 0.85 × $60 = $51.00. Sales tax of 8% makes the total 1.08 × $51.00 = $55.08. Applying the tax to the discounted price, not the original price, is the key step.
The total number of students is 12 + 18 = 30, so the ratio of girls to all students is 12 to 30. Dividing both terms by their greatest common factor of 6 gives 2 to 5. The ratio 2 to 3 would compare girls to boys rather than to the whole class.
Set up the proportion 1 inch / 12 miles = 3.5 inches / d miles. Cross-multiplying gives d = 12 × 3.5 = 42 miles. The scale factor is applied to the entire measured distance, including the half inch.
Percent change is the amount of change divided by the original amount, then multiplied by 100. The change is 50 - 40 = 10, and 10 ÷ 40 = 0.25, which is 25%. Dividing by the new value instead of the original is a common error.
Evaluate the parentheses first: 8 - 5 = 3. Then apply the exponent: 3^2 = 9. Multiply 3 × 9 = 27, and finally add 2 to get 29, following the order of operations.
Fractions can be added only when they refer to the same-sized parts, which requires a common denominator. Rewriting gives 3/6 + 2/6 = 5/6, not 2/5. Adding numerators and denominators separately treats the denominator as a count rather than as the size of each part.
An irrational number cannot be written as a ratio of two integers and has a nonterminating, nonrepeating decimal expansion. The square root of 2 is approximately 1.41421356... and never repeats or terminates, so it is irrational. The other choices are all rational.
The number of each item must be a common multiple of 6 and 8, and the least such number is the least common multiple. The multiples of 6 are 6, 12, 18, 24, ... and of 8 are 8, 16, 24, ..., so the LCM is 24. Buying 24 of each uses 4 packs of pencils and 3 packs of erasers with nothing left over.
A prime number has exactly two distinct factors, 1 and itself, and a composite number has more than two factors. The number 1 has only one factor, namely 1, so it fits neither definition. This is why 1 is classified separately as a unit rather than as prime or composite.
Multiplying by 0.25 is the same as multiplying by 1/4, since 0.25 = 25/100 = 1/4. Multiplying any number by 1/4 produces one quarter of that number. For example, 20 × 0.25 = 5, which is a quarter of 20.