Decimal representation

After the decimal point, the digits are \(9,0,9,0,\ldots\). The digit \(9\) is followed by \(0\), and this two-digit block repeats continuously; therefore, the recurring part is \(90\). The digit \(9\) alone is not the repeating block because it is followed by \(0\) each time. Exam tip: identify the shortest block after the decimal point that repeats in the same order.

The governing concept is comparison of decimal place values. Express all numbers to four decimal places: 0.8750, 0.8570, 0.8705, and 0.8700. The tenths digit is 8 in every number, so compare the hundredths digits. The value 0.8570 has 5 and is immediately smaller than the others, which have 7. Among the remaining values, compare the thousandths digits: 0.8750 has 5, whereas 0.8705 and 0.8700 have 0. Therefore 0.8750 is the greatest, so option A is correct. Notice that 0.87 is exactly 0.8700; trailing zeros do not change value. Option C is close but remains smaller than 0.875.

( \frac{1}{3}=0.333\ldots ) and ( \frac{1}{2}=0.5 ), so (0.34) lies between them. Convert boundaries to decimals for comparison.

( \frac{41}{200}=\frac{205}{1000}=0.205 ). Multiply denominator (200) by (5) to make (1000).

Digits after the decimal point have place values tenths, hundredths, thousandths, ten-thousandths, and hundred-thousandths. In 7.04004, the final 4 is the fifth digit after the decimal point, so it is in the hundred-thousandths place. Its place value is therefore 4 × 1/100000 = 4/100000. Option D is correct; the other choices belong to earlier decimal positions.

\(12.5 \times 0.1 = 1.25\). Since \(0.1 = \frac{1}{10}\), multiplying a number by \(0.1\) is equivalent to dividing it by \(10\). Therefore, \(12.5 \div 10 = 1.25\). Option B leaves the number unchanged, so it is incorrect. Exam tip: When multiplying a decimal by \(0.1\), move the decimal point one place to the left.

Multiplying both by (100) gives (36\div12=3). Multiplying both parts of a division by the same number does not change the quotient.

A non-terminating decimal without fixed repetition is non-terminating non-recurring. Such decimals can represent irrational numbers.

The number of zeros between successive 1s increases as 1, 2, 3, 4, …, so no fixed block repeats. Its decimal expansion is non-terminating and non-recurring; hence it is irrational. Exam tip: a non-terminating decimal is rational only when it eventually repeats.

In the decimal 0.312312312…, the three-digit block 312 repeats continuously. Therefore, the bar must be placed over the complete repeating block: \(0.\overline{312}\). In \(0.3\overline{12}\), only 12 is treated as repeating, which does not match the given decimal pattern. Exam tip: Before using bar notation, identify the shortest block that repeats continuously.

Compare decimal numbers from left to right. Both numbers have 0 in the ones place, but in the tenths place, x has 6 while y has 0. Since 6>0, 0.608>0.068; hence, (x>y). The relation (x<y) would be true only if the first differing decimal digit of x were smaller. Exam tip: Compare ones, then tenths, hundredths, and so on.

First reduce the fraction before applying the decimal-expansion test. The greatest common divisor of 45 and 150 is 15, so 45/150 = 3/10. The denominator 10 has only the prime factors 2 and 5, which is the exact condition for a rational number in lowest terms to have a terminating decimal expansion. Indeed, 3/10 = 0.3, which ends after one decimal place. Therefore option A is correct. Looking only at the original denominator 150 can be misleading because it contains 3, but that factor disappears during simplification. Option B would apply if a factor other than 2 or 5 remained in the reduced denominator; C is impossible for a rational number here.

( \frac{1}{9}=0.111\ldots ), so (0.125) is the greatest. Convert fractions into decimals for comparison.

(400\times25=10000), so ( \frac{27}{400}=\frac{675}{10000}=0.0675 ). Convert the denominator into a power of (10) to find the decimal.

The tenths digits are equal and the hundredths digit (a) must be greater than (3). So the smallest value of (a) is (4).

The number of zeros keeps changing, so there is no fixed recurring part. Such a decimal is non-terminating non-recurring.

The governing concept is decimal representation of a rational number. To make the denominator a power of 10, multiply numerator and denominator by 5, because 200 × 5 = 1000. Thus, 17/200 = (17 × 5)/(200 × 5) = 85/1000 = 0.085. Therefore, option A is correct. The digits must be placed according to thousandths: 85 thousandths is 0.085, not 0.85. Option B would represent 17/1000, while option C is ten times too large. Option D is greater than 1, whereas 17/200 is clearly less than 1 because the numerator is smaller than the denominator. The terminating decimal occurs because the denominator factors only into 2s and 5s.

(400\times25=10000), so ( \frac{31}{400}=\frac{775}{10000}=0.0775 ). Making the denominator a power of (10) is useful.

(0.21875=\frac{21875}{100000}=\frac{7}{32}). Convert the decimal to a fraction and reduce it to simplest form.

(1.0625=\frac{10625}{10000}=\frac{17}{16}). Keep the final improper fraction in simplest form.