Decimal representation

The correct answer is 5.75. Since 0.00999\ldots = 0.01, we get 5.74999\ldots = 5.74 + 0.00999\ldots = 5.75. The number 5.749 simply stops after three decimal places, whereas the given decimal has infinitely repeating 9s. Exam tip: when infinitely many 9s follow a digit, the decimal can be written by increasing the preceding place by 1.

After the decimal point, the places represent tenths, hundredths, thousandths and ten-thousandths, respectively. In 3.090800, 8 is the fourth digit after the decimal point, so its place value is \(\frac{8}{10000}\), or 0.0008. \(\frac{8}{1000}\) would represent the value of a digit in the third decimal place. Exam tip: Count decimal places from immediately after the decimal point.

The direct answer is option A: 0.1875=3/16<0.18̅7. First, divide 3 by 16: 3/16=0.1875 exactly. The notation 0.18̅7 means that only 7 repeats after 18, so its value is 0.187777… . Compare 0.187500… with 0.187777…: the first three decimal digits agree, but at the fourth digit 5 is less than 7, so 0.1875 is smaller. Therefore the equality and inequality in option A are both correct. Option B wrongly says 0.1875 is less than 3/16, although they are exactly equal. It also wrongly makes the recurring decimal equal to 3/16. Option C reverses the correct inequality. Option D says all three are equal, but the recurring decimal is larger. A useful check is to append zeros to a terminating decimal before comparing: 0.1875=0.187500….

Write all the numbers to four decimal places: \(b=0.3600\), \(a=0.3060\), and \(c=0.3006\). Thus, \(b\) is the greatest. Also, \(a=0.3060\) is greater than \(c=0.3006\) because at the hundredths place, \(a\) has 6 while \(c\) has 0. Therefore, the descending order is \(b>a>c\). Option \(b>c>a\) is wrong because \(0.3006<0.3060\). Exam tip: Add trailing zeros to make the decimal places equal before comparing decimals.

( \frac{11}{125}=0.088 ) and ( \frac{9}{80}=0.1125 ), so the sum is (0.2005). First convert the fractions into decimals.

In \(0.0767676\ldots\), the first digit after the decimal point, \(0\), occurs only once. After that, the block \(76\) repeats: \(0.0\,76\,76\,76\ldots\). Therefore, the bar is placed only over \(76\), giving \(0.0\overline{76}\). Option A incorrectly treats \(076\) as the repeating block. Exam tip: Before placing a bar, identify the shortest block of digits that repeats continuously.

A decimal terminates only when the denominator \(q\), in lowest form, has no prime factors other than 2 and 5; hence \(q=2^m5^n\). Any other prime factor gives a non-terminating recurring decimal. Exam tip: reduce the fraction first.

A fraction with denominator \\(2^5\times5^2\\) can be converted into a denominator that is a power of 10. To do this, the smaller power, \\(5^2\\), must be matched with two more factors of 5, while the two factors of 2 are already available. The resulting denominator is \\(2^5\times5^5=10^5\\). Hence the decimal can require at most five places.

The maximum is determined by the larger exponent, not by adding the exponents. Thus it is 5, which is option B. For example, multiplying numerator and denominator by \\(5^3\\) gives a denominator of \\(10^5\\). Some fractions may have fewer places because cancellation or trailing zeros can occur, but five is the greatest possible number under the stated denominator condition.

The number of zeros between (5)'s increases, so there is no fixed repetition. Such a decimal is non-terminating non-recurring.

To convert a decimal into a percentage, multiply it by 100: \(0.0016\times100=0.16\). Therefore, the percentage is \(0.16\%\). Option A results from shifting the decimal by only one place, whereas multiplication by 100 shifts it two places to the right. Exam tip: For decimal-to-percentage conversion, move the decimal point two places to the right and add the percent sign.

In lowest terms, if \(q\) has a prime factor other than 2 or 5, it cannot divide any \(10^n\), so the decimal is non-terminating recurring. A denominator containing only 2 and 5 gives a terminating decimal. Exam tip: reduce the fraction first.

( \frac{5}{8}=0.625 ), so (0.6245) is greater than (0.624) and less than (0.625). Convert boundary values into decimals.

(3200\times3125=10000000), so ( \frac{83}{3200}=0.0259375 ). Convert the denominator into a power of (10) to get the exact decimal.

The tenths digits are equal and the hundredths digit must satisfy (a\leq4). So (a=0,1,2,3,4), giving (5) possible values.

For a denominator of \\(2^4\times5^9\\), the powers of 2 and 5 must be balanced to form a power of 10. There are only four factors of 2 but nine factors of 5, so multiply by \\(2^5\\) to obtain \\(2^9\times5^9=10^9\\). Therefore a terminating decimal can have at most nine decimal places.

The rule is that the maximum number of places is the larger of the exponents of 2 and 5 in the simplified denominator. Here, \\(\max(4,9)=9\\), so option C is correct. The exponents should not be added to get 13, because the factors are paired to make tens, with each pair of 2 and 5 producing one factor of 10.

(1250\times8=10000), so ( \frac{73}{1250}=\frac{584}{10000}=0.0584 ). Making the denominator a power of (10) is an exact method.

(0.00015625=\frac{15625}{100000000}=\frac{1}{6400}). First count decimal places and then simplify the fraction.

The denominator has (13), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

The decimal digits are 3, 4, 0, 9, 0, 9, \(\ldots\). After 34, the block 09 repeats continuously, so the correct notation is \(0.34\overline{09}\). In \(0.\overline{3409}\), the entire block 3409 is incorrectly treated as repeating from the beginning. Exam tip: Before placing a bar, identify the smallest block of digits that repeats indefinitely.

The direct answer is A: \(2.07363636\ldots\). The notation \(2.07\overline{36}\) means that the digits 07 occur once after the decimal point and only the block 36 is repeated. Therefore write 2, then 07, then 36 again and again: \(2.07363636\ldots\). Option A is exactly this expansion. Option B, \(2.0736736\ldots\), changes the repeating order and inserts an incorrect digit. Option C, \(2.07073636\ldots\), repeats or inserts 07 again, although the bar is not over 07. Option D, 2.0736, stops after one copy and is therefore not the full recurring decimal. The bar tells you precisely which digits repeat; it does not mean that every digit after the decimal repeats. Memory cue: first write the digits outside the bar once, then repeat only the digits under the bar forever.

For \(\frac{7}{48}\), the denominator is \(48=2^4\times3\). A fraction in simplest form has a non-terminating recurring decimal when its denominator contains a prime factor other than 2 or 5. The other denominators contain only 2 and/or 5. Exam tip: simplify first, then factorise the denominator.

( \frac{17}{24}=0.70833\ldots ), which is less than (0.7084). View all numbers in decimal form for comparison.

The number of zeros between (9)'s keeps changing, so there is no fixed recurring part. Such a decimal is non-terminating non-recurring.

(256\times390625=100000000), so ( \frac{23}{256}=0.08984375 ). A denominator that is a power of (2) gives a terminating decimal.

(0.0021875=\frac{21875}{10000000}=\frac{7}{3200}). Count decimal places and simplify the final fraction.