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 correct answer is option A: \(5.209363636\ldots\). The notation \(5.209\overline{36}\) means that the digits before the bar, 209, occur once and the digits under the bar, 36, repeat forever. Thus the decimal is written as 5.20936363636... The first digits are 5.209, followed by 36, then 36 again, and so on. Option A exactly shows this pattern. Option B, 5.20936209..., wrongly inserts 209 again after 36; the non-repeating part must not restart. Option C, 5.20936, stops after one copy of 36, so it is not the complete recurring decimal. Option D, 5.20920936..., repeats or inserts 209 in the wrong place. The bar tells us precisely which digits continue indefinitely; it does not apply to all digits. Memory cue: write the digits before the bar once, then repeat only the barred block.

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 direct answer is option B, 7. Compare the decimals from left to right. In 0.94a7 and 0.9467, the digits before the third decimal place are the same: 9 is in the tenths place and 4 is in the hundredths place. Therefore, the first deciding digit is the thousandths digit. In the first number it is a, while in the second number it is 6. For the first number to be greater, a must be greater than 6. The digits greater than 6 are 7, 8, and 9, so the smallest possible value is 7. Option A, 5, is wrong because 0.9457 is less than 0.9467. Option B, 7, is correct because 0.9477 is greater than 0.9467. Option C, 6, is wrong because the numbers then become 0.9467 and 0.9467, so they are equal, not greater. Option D, 8, would also make the inequality true, but it is not the smallest possible value. Remember: when decimal places to the left are equal, compare the first place where they differ.

(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 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.

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.

The correct answer is option B: \(0.000078125\). Start with \(100000x=7.8125\). To isolate x, divide both sides by 100000: \(x=\frac{7.8125}{100000}\). Since 100000 has five zeros, move the decimal point five places to the left: 7.8125 becomes 0.000078125. Therefore B is correct. Option A, 781250, comes from multiplying or moving the decimal in the wrong direction. Option C, 0.0078125, moves the decimal only three places, so it represents division by 1000 rather than 100000. Option D, 0.078125, moves it only two places, so it represents division by 100 rather than 100000. A good check is to multiply the answer by 100000: \(0.000078125\times100000=7.8125\). Remember: dividing by a power of 10 moves the decimal left; the number of places equals the number of zeros.

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.

To find the multiplier, divide the larger decimal by the smaller one: \(0.0078125 \div 0.000078125 = 100\). Therefore, \(0.0078125\) is 100 times \(0.000078125\). Exam tip: for “how many times,” divide the compared larger quantity by the reference quantity.

The bar is placed only over 54, so the block 54 repeats indefinitely. The first three zeros are non-repeating; hence the expansion is \(0.000545454\ldots\). Option C incorrectly treats 540 as the repeating block. Exam tip: first identify exactly which digits are covered by the bar in a recurring decimal.

The number of zeros between successive 1s is 1, 2, 3, 4, …, so no fixed repeating block exists. Hence the decimal is non-terminating and non-repeating, making it irrational. Exam tip: check repetition, not just the digits used.

First reduce the fraction before deciding the type of decimal expansion. The numerator and denominator have a common factor 44: \\(\frac{132}{484}=\frac{3}{11}\\). The simplified denominator is 11, which is a prime factor other than 2 or 5. A rational number whose simplified denominator contains such a factor has a non-terminating recurring decimal expansion.

For confirmation, division gives \\(\frac{3}{11}=0.272727\ldots\\), where the block 27 repeats endlessly. Thus option C is correct. It is important not to judge the original denominator 484 without simplification, because common factors may disappear. Here, however, the remaining factor 11 clearly prevents termination and produces repetition.

Here, 23.74999\ldots = 23.74 + 0.00999\ldots. Since 0.00999\ldots = 0.01, the value is 23.74 + 0.01 = 23.75. The number 23.749 has only finitely many digits and does not include the effect of the infinitely repeating 9s. Exam tip: An infinite string of 9s after a decimal place can be written by increasing the preceding digit by 1.

In 8.090600, the digits after the decimal point represent tenths, hundredths, thousandths and ten-thousandths in order. The digit 6 is in the fourth place after the decimal point, so its place value is \(6 \times \frac{1}{10000}=\frac{6}{10000}\). Therefore, option C is correct. Exam tip: count decimal places from left to right; the fourth place always has denominator 10,000.

Writing the numbers up to four decimal places gives \(b=0.7800\), \(a=0.7080\), and \(c=0.7008\). Thus, \(0.7800>0.7080>0.7008\), so the correct order is \(b>a>c\). Between \(a\) and \(c\), the thousandths digit of 0.7080 is 8, while that of 0.7008 is 0; hence \(a>c\). Exam tip: Append zeros to decimals when needed so that all numbers have the same number of decimal places before comparing them.

( \frac{47}{500}=0.094 ) and ( \frac{23}{1280}=0.01796875 ), so the sum is (0.11196875). First convert the fractions into decimals.