Decimal representation

We can write 0.4 as 0.40 because adding a zero at the end of a decimal does not change its value. Therefore, 0.36 + 0.40 = 0.76. Option 0.396 incorrectly places digits together instead of adding according to place value. Exam tip: Always align the decimal points before adding decimal numbers.

0.5 can be written as 0.50 because adding a zero to the right of a decimal does not change its value. Thus, 1.25 - 0.50 = 0.75. The option 0.80 may result from incorrect subtraction of decimal digits. Exam tip: Align the decimal points before subtracting decimal numbers.

The direct answer is option D: 0.05 is greater. To compare decimals, first compare the whole-number parts; both are 0. Then compare tenths, hundredths, and thousandths from left to right. Add a zero at the end without changing a decimal's value: 0.05 = 0.050. Now compare 0.050 and 0.005. At the tenths place both have 0, but at the hundredths place 0.050 has 5 while 0.005 has 0, so 0.050 is larger. Option A, 0.005, is the smaller number, not the greater one. Option B, both equal, is wrong because their hundredths digits differ. Option C, cannot be compared, is wrong because all terminating decimals can be compared by lining up their places. Option D is correct. Remember: trailing zeros do not change value, but the position of a nonzero digit matters greatly.

The whole-number part of a decimal is the part before the decimal point. In 15.08, the decimal point separates 15 from 08. Therefore the whole-number part is 15, making choice A correct. The digits 0 and 8 are after the decimal point and describe the fractional part, which is 0.08 or eight hundredths.

Place value makes the distinction clear: 1 is in the tens place and 5 is in the ones place, so they form the integer 15. The zero in the tenths place and the 8 in the hundredths place together form the decimal portion. Writing 08 alone does not identify the whole-number part, and writing 1508 would remove the decimal point and change the value. Thus 15 is the required answer.

The governing place-value concept is that the whole-number part lies to the left of the decimal point, while the decimal part is written to its right. In 15.08, 15 is the whole-number part and 08 is the decimal part as displayed, so option B is correct. The zero in 08 should not be casually removed when identifying the written decimal part because it indicates zero tenths and eight hundredths. Numerically, 15.08 may also be written as 15.080 without changing its value. Option A is the integer part, option C is formed by deleting the decimal point, and option D is a different number created by rearranging digits. None of those represents the decimal part of the given number.

The decimal expansion of an irrational number neither terminates nor repeats. So it is non-terminating non-recurring.

The decimal expansion of a rational number is terminating or non-terminating recurring. This identification is very useful in exams.

In 6.999..., the digit after the decimal point is 9, and it continues as 9, 9, 9, ... indefinitely. Therefore, the repeating digit is 9. Option 99 is a two-digit group, whereas the question asks for a single digit. Exam tip: Identify the digit or group of digits that repeats continuously after the decimal point.

After the decimal point, the digits repeat in the group 0, 3, 4: 034 | 034 | 034 | \ldots. Therefore, the recurring part is 034. It cannot be only 34 because a 0 occurs before every repetition of 34. Exam tip: Split the digits after the decimal point into repeated equal blocks to identify the recurring part.

The direct answer is B: 0.031. A denominator of 1000 means the numerator is divided into thousandths. Since \(1000=10^3\), move the decimal point three places to the left in 31: 31 becomes 0.031. Equivalently, write 31 with three digits after the decimal point: 031, giving 0.031. Option A, 0.31, represents \(31/100\), not \(31/1000\). Option B is correct. Option C, 0.0031, has four decimal places and represents \(31/10000\). Option D, 3.1, is much larger and does not represent a fraction less than 1. The useful place-value rule is: denominator 10, 100, or 1000 requires one, two, or three decimal places respectively.

To arrange decimals in ascending order means to place them from the smallest value to the greatest value. Decimal places must be compared from left to right, and missing digits may be written as zeros. Thus 0.9 can be written as 0.900, 0.09 as 0.090, and 0.009 already has three decimal places. This makes comparison clear.

All numbers have the same whole part, 0. Compare the tenths digits first: 0.009 and 0.09 have tenths digit 0, while 0.9 has tenths digit 9, so 0.9 is greatest. Between the first two, compare hundredths: 0.009 has 0 hundredths, whereas 0.090 has 9 hundredths. Hence 0.009<0.09<0.9, which is choice B. Extra zeros do not change a decimal's value.

( \frac{37}{200}=\frac{185}{1000}=0.185 ). Converting the denominator into (1000) helps find the decimal quickly.

The governing concept is conversion of a rational number into decimal notation. Since the denominator is 40, multiply numerator and denominator by 25 to create a denominator of 1000: 7/40 = (7 × 25)/(40 × 25) = 175/1000. A fraction with denominator 1000 represents 175 thousandths, which is 0.175. Therefore option A is correct. Long division gives the same result: 7 divided by 40 equals 0.175. The value 0.740 incorrectly treats 7 and 40 as if they formed a decimal, while 0.075 would represent 3/40, not 7/40. The value 1.75 is greater than 1, whereas 7/40 is less than 1 because the numerator is smaller than the denominator. The denominator's factors are only 2 and 5, so the decimal terminates rather than repeating.

The direct answer is C: \(\frac{27}{1000}\). The decimal 0.027 has three digits after the decimal point, so it means 27 thousandths: \(0.027=\frac{27}{1000}\). The numerator 27 and denominator 1000 have no common factor greater than 1, because 27 has factors 1, 3, 9, 27, while 1000 has only factors 2 and 5. Thus the fraction is already simplified. Option A, \(27/100\), equals 0.27 and is ten times too large. Option B, \(27/10\), equals 2.7 and is much too large. Option C is correct. Option D, \(27/10000\), equals 0.0027 and is ten times too small. Count decimal places carefully: three places require denominator 1000.

In 6.214, the digit 1 is in the second place to the right of the decimal point. Therefore, its place value is one hundredth, \(\frac{1}{100}\). The first decimal place is tenths, \(\frac{1}{10}\), so option A is incorrect. Exam tip: After the decimal point, the places are tenths, hundredths, and thousandths in order.

In 3.209, 3 is in the ones place, 2 is in the tenths place, and 9 is in the thousandths place. Therefore, its expanded form is \(3+\frac{2}{10}+\frac{9}{1000}\). In option B, the place values of 2 and 9 are interchanged, so it is incorrect. Exam tip: the first, second, and third digits after the decimal represent tenths, hundredths, and thousandths, respectively.

The direct answer is C: 4.065. To compare correctly, write both endpoints with three decimal places: 4.06 = 4.060 and 4.07 = 4.070. A number between them must be greater than 4.060 and less than 4.070. Option C, 4.065, satisfies both inequalities: \(4.060<4.065<4.070\). Option A, 4.059, is less than 4.060, so it is outside the interval. Option B, 4.070, equals the upper endpoint, so it is not strictly between the two numbers. Option D, 4.075, is greater than 4.070, so it is outside. Adding trailing zeros helps compare decimal places without changing a number. The word “between” usually means strictly greater and strictly smaller than the endpoints.

After the decimal point, the digits repeat in the order 456, 456, 456. Therefore, the recurring part is 456. The groups 45 and 56 are only parts of the pattern and are not repeated as complete blocks. Exam tip: identify the smallest group of digits that repeats continuously without any change.

(45=3^2\times5) has (3), so the decimal will not terminate. Since it is rational, it will be non-terminating recurring.

The governing concept is comparison of decimal numbers by aligning place values. Write the numbers with four decimal places: 0.1250, 0.1520, and 0.1025. Their whole-number parts and tenths digits are the same, so compare the hundredths digits. The hundredths digits are 2, 5, and 0 respectively; the largest is 5, belonging to 0.1520. Thus 0.152 > 0.125 and 0.152 > 0.1025, so option B is correct. The numbers are not equal because their digits differ at the hundredths place. Adding a trailing zero to 0.125 does not change its value; it only makes the lengths equal for comparison. Option C is actually the smallest of the three, while option A is between the other two. The digit-by-digit comparison confirms the answer without requiring conversion to fractions.

In a decimal number, each digit has a value determined by its place. In 2.045, the digit 2 is in the units place, 0 is in the tenths place, 4 is in the hundredths place, and 5 is in the thousandths place. Hence the place-value expansion is 2 + 0/10 + 4/100 + 5/1000, which is equal to 2 + 4/100 + 5/1000. Therefore option A is correct. Option B incorrectly treats 4 as tenths and 5 as hundredths. Option C combines 4 and 5 as 45 hundredths, losing the fact that 4 is hundredths and 5 is thousandths. Option D changes the whole-number part from 2 to 20.

The governing concept is place value when multiplying by a power of ten. Since 100=10², multiplication by 100 shifts the decimal point two places to the right. Starting with 5.08, the first shift gives 50.8 and the second gives 508; therefore 5.08×100=508. Option C is correct. Option A is smaller and corresponds to moving the decimal point left, as in division by 10. Option B represents multiplication by only 10, because it shifts the point one place. Option D shifts it three places and would correspond to multiplication by 1000. The same result can be checked by writing 5.08=508/100, so (508/100)×100=508.

The governing concept is decimal place value under division by a power of ten. Because 1000=10³, dividing by 1000 moves the decimal point three places to the left. Starting from 37.5, the first shift gives 3.75, the second gives 0.375, and the third gives 0.0375. Hence 37.5÷1000=0.0375, so option A is correct. Option B represents division by 100, involving only two leftward shifts. Option C represents division by 10, involving one shift. Option D moves the decimal point in the opposite direction and is larger than the original positive number; division by 1000 cannot produce such a result. The calculation can also be checked as 37.5/1000=375/10000=0.0375.

The governing concept is decimal division. To remove the decimal points without changing the quotient, multiply both the dividend and divisor by 1000, because each has at most three decimal places. Thus, 0.625 ÷ 0.025 = 625 ÷ 25. Now 25 × 25 = 625, so 625 ÷ 25 = 25. Therefore, option B is correct. Option A, 2.5, would result from an incorrect placement of the decimal point. Option C, 250, is ten times too large, while option D, 0.25, is much too small. Multiplying both numbers by the same nonzero number preserves their ratio, which is why this method is valid.

The governing algebraic idea is to isolate the variable by performing the inverse operation on both sides. Since 10000 is multiplying x, divide both sides by 10000: x = 0.5625/10000. Dividing by 10,000 moves the decimal point four places to the left. Starting with 0.5625, the successive shifts give 0.05625, 0.005625, 0.0005625, and finally 0.00005625. Hence x = 0.00005625, so option C is correct. Option A ignores the division, option B shifts the decimal only three places, and option D shifts it only two places. Substitution confirms the result because 10,000 × 0.00005625 = 0.5625.