Decimal Fractions Questions and Answers

Answer: A

The expression is set up to be the square of a sum. Comparing with

\[ (a+b)^{2} = a^{2} + 2ab + b^{2} \]

the outer terms give \( a = 11.98 \) and \( b = 0.02 \), so the middle term must be

\[ 2ab = 2(11.98)(0.02) = 0.4792 \]

The middle term as written is \( 11.98x \), so

\[ 11.98x = 0.4792 \quad\Longrightarrow\quad x = 2(0.02) = 0.04 \]

The square is then \( (11.98 + 0.02)^{2} = 12^{2} = 144 \). Notice the shortcut: \( x \) is simply twice the smaller number, so the decimals never need multiplying out.

Answer: A

The unknown is being subtracted, so it equals everything on the left minus the result on the right.

\[ 4211.01 + 22.261 - x = 2645.759 \]

\[ x = 4211.01 + 22.261 - 2645.759 \]

\[ 4211.010 + 22.261 = 4233.271 \]

\[ 4233.271 - 2645.759 = 1587.512 \]

The three near-identical distractors differ only in the last three digits, so the subtraction has to be carried out to full precision rather than estimated.

Answer: C

Multiply as whole numbers, then place the decimal point using the total number of decimal places in the factors.

(i) \( 261 \times 13 = 3393 \); two places plus one is three, so

\[ 2.61 \times 1.3 = 3.393 \]

(ii) \( 21693 \times 14 = 303702 \); four places plus one is five, so

\[ 2.1693 \times 1.4 = 3.03702 \]

(iii) \( 4 \times 4 \times 4 \times 40 = 2560 \); one, two and three places with none from the 40 gives six, so

\[ 0.4 \times 0.04 \times 0.004 \times 40 = 0.002560 \]

Answer: B

Rearrange each equation for the unknown.

(i) From \( 0.006 \div ? = 0.6 \):

\[ ? = \frac{0.006}{0.6} = 0.01 \]

(ii) From \( ? \div 0.025 = 80 \):

\[ ? = 80 \times 0.025 = 2 \]

The two rearrange in opposite directions: in the first the unknown is the divisor and so divides across, in the second it is the dividend and so multiplies.

Answer: B

Isolate the question mark in each equation.

(i)

\[ ? = 9318.678 - 5172.49 - 378.352 = 3767.836 \]

(ii)

\[ ? = 5169.38 + 7328.96 = 12498.34 \]

Note the direction of each operation: the first needs subtraction because the unknown sits on the adding side, and the second needs addition because it is being subtracted from.

Answer: C

\(\dfrac {29.94}{1.45}\\~\\= \dfrac {299.4}{14.5} \\~\\ =\left ( \dfrac {2994}{14.5} \times \dfrac {1}{10} \right )\\~\\=\dfrac {172}{10} \left [ \because \dfrac {2994}{14.5} = 172 \right ]\\~\\= 17.2\)

Answer: D

\(4.\overline{86} = \left(4 + \dfrac{86}{99}\right)\)
\(3.\overline{71} = \left(3 + \dfrac{71}{99}\right)\)
Hence,
\(4.\overline{86} - 3.\overline{71} \\~\\= \left(4 + \dfrac{86}{99}\right) - \left(3 + \dfrac{71}{99}\right)\\~\\ = \left( 1 + \dfrac{15}{99}\right)\\~\\ = 1.\overline{15}\)

Answer: D

Work left to right, treating the subtraction as a negative term:

\[ 945.341 - 1042.792 = -97.451 \]

\[ -97.451 + 875.435 = 777.984 \]

\[ 777.984 + 31.025 = 809.009 \]

Grouping the positives first gives the same result: \( 945.341 + 875.435 + 31.025 = 1851.801 \) and \( 1851.801 - 1042.792 = 809.009 \).

Answer: C

Shifting the decimal point moves both numbers by a known power of ten, so the answer follows from the one given without redoing the division.

\[ \frac{29.94}{1.45} = \frac{2994 \div 100}{14.5 \div 10} = \frac{2994}{14.5} \times \frac{10}{100} = 172 \times \frac{1}{10} = 17.2 \]

The numerator shrank by a factor of 100 and the denominator by only 10, so the quotient shrinks by the ratio of those, a factor of 10.

Answer: D

Clear the decimal from the divisor by multiplying both numbers by the same power of ten, then divide.

(i) \[ \frac{35}{0.07} = \frac{3500}{7} = 500 \]

(ii) \[ \frac{2.5}{0.0005} = \frac{25000}{5} = 5000 \]

(iii) \[ \frac{136.09}{43.9} = \frac{1360.9}{439} \approx 3.1 \]

Dividing by a number smaller than 1 always makes the result larger, which is why the first two answers are so much bigger than the numbers they start from.