Decimal Fractions Questions and Answers

Answer: C

Write each decimal over the power of ten its last digit sits on, then cancel.

(i) \[ 0.75 = \frac{75}{100} = \frac{3}{4} \]

(ii) \[ 3.004 = \frac{3004}{1000} = \frac{751}{250} \]

(iii) \[ 0.0056 = \frac{56}{10000} = \frac{7}{1250} \]

The denominator is always \( 10^{n} \) for \( n \) decimal places, and the cancelling divides by 25, 4 and 8 respectively.

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: D

The three terms are the expansion of a squared difference, so no long multiplication is needed:

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

With \( a = 3.75 \) and \( b = 2.75 \),

\[ (3.75 - 2.75)^{2} = 1^{2} = 1 \]

Multiplying the three products out gives 14.0625, 20.625 and 7.5625, and \( 14.0625 - 20.625 + 7.5625 = 1 \) - the same answer for considerably more work.

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: 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.

Answer: A

Put everything over a common denominator of 24 and compare:

\[ \frac{1}{3} = \frac{8}{24}, \qquad \frac{7}{8} = \frac{21}{24} \]

So the fraction must lie strictly between \( \tfrac{8}{24} \) and \( \tfrac{21}{24} \).

• \( \tfrac{17}{24} \) - between them ✓
• \( \tfrac{23}{24} \) - above \( \tfrac{21}{24} \)
• \( \tfrac{11}{12} = \tfrac{22}{24} \) - also above
• \( \tfrac{1}{4} = \tfrac{6}{24} \) - below \( \tfrac{8}{24} \)

As decimals the answer is 0.708, comfortably between 0.333 and 0.875.

Answer: A

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

Multiplying by a power of ten moves the decimal point right by as many places as the power has zeros.

(i) \[ 6.3204 \times 100 = 632.04 \]

(ii) \[ 0.069 \times 10000 = 690 \]

In the second the point moves four places but only three digits are available, so a zero is written in to fill the gap: 0.069 becomes 0690, that is 690.

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: 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.