Decimal representation

(2) is in the first place after the decimal, so its place value is ( \frac{2}{10} ). The first decimal place is called tenths.

A zero at the extreme right of a decimal does not change its value. Hence, \(5.060=5.06\). In 5.006, the digit 6 is in the thousandths place, whereas in 5.060 it is in the hundredths place. Exam tip: Only trailing zeros may be removed; zeros between digits can affect place value.

( \frac{13}{50}=\frac{26}{100}=0.26 ). Making the denominator (100) helps find the decimal easily.

In 0.0909..., the two-digit block 09 repeats continuously. Therefore, the bar must be placed over the complete repeating block: \(0.\overline{09}\). Option \(0.0\overline{9}\) shows only 9 as recurring, so it does not represent the given repeating pattern correctly. Exam tip: First identify the shortest block of digits that repeats.

An irrational number has a non-terminating non-recurring decimal expansion. A non-terminating decimal without fixed repetition can be irrational.

In (0.875), the digits after the decimal are (8), (7), and (5), so there are (3) digits. Count zeros too when they occur after the decimal.

In \(7.003\), the part to the right of the decimal point is \(.003\). Therefore, the decimal part is \(.003\). \(7\) is the integer part, whereas \(003\) are the digits after the decimal point. Exam tip: To identify the decimal part, look to the right of the decimal point and write its value with the decimal point.

A decimal number has a whole-number part to the left of the decimal point and a fractional part to the right. In 12.45, the decimal point separates 12 from 45. Therefore 12 is the whole-number part, and 45 represents forty-five hundredths, or 0.45. Hence choice C is correct.

This can also be understood by place value. The digit 1 is in the tens place and 2 is in the ones place, so together they form the whole number 12. The digits 4 and 5 lie after the decimal point and form the fractional part. The number 1245 would ignore the decimal point, while 45 alone is only the digits after it. Thus the complete number is 12+0.45, and its whole-number part is 12.

Write 0.09 as 0.090 to compare the decimals. In 0.091, the thousandths digit is 1, whereas in 0.090 it is 0. Therefore, 0.091 is greater than 0.09. Note that 0.090 is equal to 0.09 because a zero added at the end of a decimal does not change its value. Exam tip: Add zeros to make the number of decimal places equal before comparing decimals.

The governing concept is comparison of decimal numbers using place value and the number line. Rewrite the endpoints with equal decimal places: 0.3 = 0.30 and 0.4 = 0.40. A number lies between them when it is greater than 0.30 and less than 0.40. The number 0.35 satisfies 0.30 < 0.35 < 0.40, so option C is correct. The value 0.25 is less than 0.30, while 0.45 and 0.50 are greater than 0.40. On a number line, 0.35 is located halfway between 0.3 and 0.4, which gives a visual confirmation. Equalizing decimal places avoids the common mistake of comparing digits without considering their place values.

(0.6\times100=60), so we get (60%). To convert a decimal into a percentage, multiply by (100).

Writing 1.25 as hundredths gives \(1.25=\frac{125}{100}\). Dividing the numerator and denominator by 25 simplifies this to \(\frac{5}{4}\), so option A is correct. Option B equals 12.5, option C equals 0.25, and option D equals 0.8. Exam tip: when a decimal has two digits after the decimal point, first write it over 100 and then reduce the fraction.

The decimal 0.25 ends after two decimal places, so it is a terminating decimal. Hence, it is not a non-terminating recurring decimal. In contrast, 0.121212\ldots, 0.777\ldots, and 3.454545\ldots repeat a digit or block of digits indefinitely. Exam tip: A decimal with \ldots and a repeating fixed pattern is non-terminating recurring.

To the right of the decimal point, the places are tenths, hundredths, thousandths, and ten-thousandths in order. In 0.0005, 5 is the fourth digit after the decimal point, so it is in the ten-thousandths place. The thousandths place is the third place after the decimal point. Exam tip: Count the digits after the decimal point to identify the place value.

For \(\frac{7}{12}\), \(12=2^2\times3\). Since 3 occurs in the denominator, its decimal form is \(0.58\overline{3}\), which is non-terminating recurring. The other denominators contain only 2 and 5. Exam tip: first reduce the fraction to lowest terms.

Dividing \(7\) by \(12\) gives \(0.58333\ldots=0.58\overline{3}\); since 3 repeats, the decimal is non-terminating recurring. \(0.58\) is only a truncated value. Exam tip: a repeated remainder produces repeating digits.

The correct sum is \(9.99+0.01=10.00\). Here, 9 hundredths plus 1 hundredth makes 10 hundredths, producing a carry of 1 and giving 10. Option \(9.991\) is incorrect because adding \(0.01\) does not add a digit in the thousandths place. Exam tip: While adding decimals, align the decimal points so that digits with the same place value are added together.

( \frac{1}{16}=\frac{625}{10000}=0.0625 ). Try to convert the denominator into (10), (100), (1000), or (10000).

For \(\frac{7}{24}\), \(24=2^3\times3\). A reduced denominator containing a prime other than 2 or 5 gives a non-terminating recurring decimal. \(\frac{3}{40}\) terminates. Exam tip: factor the denominator.

In 2.07, 2 is in the ones place and 7 is in the hundredths place. Therefore, its expanded form is \(2+\frac{7}{100}\). Option A places 7 in the tenths place, while option D equals 2.70 and option C equals 2.007. Exam tip: the first digit after the decimal point represents tenths, and the second represents hundredths.