Decimal Fractions Questions and Answers

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.

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: 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: 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.