Decimal representation

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 denominator is (1000), so (31) is written in three decimal places. Therefore, ( \frac{31}{1000}=0.031 ).

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.

(0.027=\frac{27}{1000}) and it is already in simplest form. For three decimal places, take 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.

(4.065) is greater than (4.060) and less than (4.070). Make decimal places equal while finding a number between two decimals.

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.

The governing concept is expressing a fraction as a terminating decimal. To make the denominator a power of ten, multiply 40 by 25, giving 1000. The numerator must also be multiplied by 25: 7/40 = (7 × 25)/(40 × 25) = 175/1000 = 0.175. Therefore option A is correct. The result is sensible because 7/40 is less than 1, so its decimal must be less than 1; this rules out 1.75 immediately. The value 0.075 equals 3/40, so it uses the wrong numerator. The value 0.740 is not the result of the fraction calculation and equals 0.74. Since 40 = 2³ × 5, its denominator contains no prime factors other than 2 and 5; consequently the decimal expansion terminates after a finite number of places. Both denominator scaling and ordinary division verify 0.175.

(80\times125=10000), so ( \frac{19}{80}=\frac{2375}{10000}=0.2375 ). Make the denominator a power of (10).

Since 0.3125 has four digits after the decimal point, it can be written as \(\frac{3125}{10000}\). Dividing the numerator and denominator by 625 gives \(\frac{3125}{10000}=\frac{5}{16}\). Option A, \(\frac{31}{100}\), and option B, \(\frac{3125}{1000}\), are not equal to the given decimal. Exam tip: use \(10^n\) as the denominator when there are n decimal places, then reduce the fraction.

The fraction 13/150 is already in simplest form because 13 shares no common factor with 150. Factor the denominator: 150 = 2 × 3 × 5². A rational number has a terminating decimal only when, after simplification, its denominator contains no prime factors other than 2 and 5. Here the factor 3 remains, so division by 150 cannot end; instead, the remainders eventually repeat and produce a recurring block. Therefore the decimal expansion is non-terminating and recurring, making option D correct. Option A would apply to a denominator made only from 2s and 5s, option B describes an irrational decimal, and option C is wrong because the fraction is not an integer.