Decimal representation

After 67, the digits 08, 08, 08, \ldots repeat. Thus, 67 is the non-repeating part and 08 is the repeating block, so the notation is \(0.67\overline{08}\). In \(0.670\overline{8}\), only 8 repeats, so it does not show the repeated zeros. Exam tip: Write the decimal digits in groups first and identify the shortest block that repeats.

The bar is only over (36), so after (209), (36) repeats. The barred part repeats continuously.

Let \(x=0.999\ldots\). Then \(10x=9.999\ldots\); subtracting gives \(9x=9\), hence \(x=1\). Finite truncations are smaller, but their limit is \(1\). Exam tip: use the \(10x-x\) method for recurring decimals.

( \frac{9}{11}=0.818181\ldots ), which is less than (0.8189). View all numbers in decimal form for comparison.

The tenths and hundredths digits are equal, so the thousandths digit must satisfy (a>6). The smallest value is (7).

(2048) is a power of (2), so the decimal terminates and ( \frac{97}{2048}=0.04736328125 ). Convert a power-of-(2) denominator into a power of (10).

(0.000546875=\frac{546875}{1000000000}=\frac{7}{12800}). Always write the final answer in simplest form.

The fraction 61/90 is already in lowest terms because 61 is prime and does not divide 90. Factor the denominator: 90 = 2 × 3² × 5. A reduced rational fraction has a terminating decimal only when its denominator contains no primes other than 2 and 5. Since 3 remains as a factor, the decimal cannot terminate; because every rational number has either a terminating or an eventually repeating decimal, it must be non-terminating recurring. Thus option B is correct. Option A ignores the factor 3, option C describes an irrational decimal, and option D is not a valid classification for this fraction.

The first (5) is in tenths and the last (5) is in millionths. Zero-value places need not be written separately.

Multiplying both numbers by (10000) gives (8640\div18=480). Removing decimals makes division easier.

Use decimal place value by writing both numbers with the same number of decimal places: 15.009 = 15.00900. Now subtract 8.34575 from 15.00900 in aligned columns. The calculation gives 15.00900 − 8.34575 = 6.66325. A reverse check confirms the result because 8.34575 + 6.66325 = 15.00900. Therefore option A is correct. Option B differs by 0.01000 and indicates a borrowing error; option C is the complementary-looking result from mishandling the subtraction; and option D reflects an error in one decimal place. Keeping the decimal points aligned prevents these mistakes.

Writing equal decimal places gives (0.5625000000+0.0019531250+0.0000390625=0.5644921875). Align decimal points while adding.

In 19.874999\ldots, infinitely many 9s occur after 4. Since \(0.004999\ldots=0.005\), we get \(19.874999\ldots=19.875\). Option 19.874 is only a truncation of the decimal, so it is not equal to the given number. Exam tip: when infinitely many 9s follow a digit, rewrite the decimal by increasing the preceding finite part appropriately.

( \frac{41}{80}=0.5125 ) and ( \frac{19}{37}=0.513513\ldots ), so (0.5130) lies between them. Convert the boundaries into decimals.

\(77\) and \(480\) are coprime, and \(480=2^5\times3\times5\). Factor 3 makes the decimal non-terminating recurring; an even denominator alone is insufficient. Exam tip: factorise the lowest denominator.

A rational number has a decimal expansion that either terminates or continues with a fixed repeating block. In 6.03003000300003…, the groups of zeros between successive 3s keep increasing, so there is no fixed block that repeats forever. The decimal is non-terminating and non-recurring, which is the characteristic decimal form of an irrational number. Therefore option C is correct. Option A is unsuitable because the decimal does not end. Option B is unsuitable because no constant cycle repeats. Option D is also impossible because an integer has no nonzero digits after the decimal point.

(x=7.8125\div100000=0.000078125). On dividing by (100000), the decimal moves five places left.

The places after the decimal point are tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths and millionths. In 43.009008, 8 is in the sixth place after the decimal point, so its place value is \(\frac{8}{1000000}\). Option C represents the fifth place and is therefore incorrect. Exam tip: Count zeros after the decimal point while determining the digit's position.

Write the numbers with equal decimal places: \(5.0050,\ 5.0055,\ 5.0505,\ 5.5000\). In \(5.0050\) and \(5.0055\), the thousandths digit is 5, but the next digit is 0 and 5 respectively; hence \(5.005<5.0055\). Also, \(5.0505\) is greater than \(5.0055\), while \(5.500\) is the greatest. Therefore, option A is correct. Exam tip: Add zeros to the right of decimals to make the number of decimal places equal before comparing.

(32000\times3125=100000000), so ( \frac{181}{32000}=0.00565625 ). Convert the denominator into a power of (10) to get the exact decimal.