Number System Questions and Answers

Answer: A

This is the closure property.

A set is closed under an operation when performing the operation on any members of the set always gives a member of the same set. Adding any two whole numbers always gives a whole number, so whole numbers are closed under addition. They are not closed under subtraction, since 3 - 5 is not a whole number.

The commutative property is about order (a + b = b + a), the associative property about grouping ((a + b) + c = a + (b + c)), and the distributive property links multiplication with addition (a(b + c) = ab + ac). None of them is what the statement describes.

Answer: A

'Three' is used in the ordinal sense in I live on the third floor of this building.

An ordinal number tells position or order in a sequence: first, second, third. 'Third floor' says which floor in order, not how many floors there are.

In the other sentences three is a cardinal number, telling how many: three rooms, three team members, sets of three pencils. Each answers 'how many?', not 'which one in order?'.

Answer: A

The largest and second largest are 5/6 and 3/4.

Write all five fractions over a common denominator; the LCM of 6, 4, 2, 3 and 5 is 60:

\[ \begin{aligned} \tfrac{5}{6} &= \tfrac{50}{60} \\ \tfrac{3}{4} &= \tfrac{45}{60} \\ \tfrac{2}{3} &= \tfrac{40}{60} \\ \tfrac{3}{5} &= \tfrac{36}{60} \\ \tfrac{1}{2} &= \tfrac{30}{60} \end{aligned} \]

In order, \( \tfrac{5}{6} \gt \tfrac{3}{4} \gt \tfrac{2}{3} \gt \tfrac{3}{5} \gt \tfrac{1}{2} \). Option 5/6 and 3/5 skips 3/4 and 2/3; 3/5 and 2/3 are the fourth and third; 3/4 and 1/2 are the second and the smallest.

Answer: D

The number that does not lie between a and b is \(-\frac{7}{9}\).

Here \(a = -\frac{3}{4} = -0.75\) and \(b = \frac{5}{6} \approx 0.83\), so a number lies between them only if it is greater than \(-0.75\) and less than 0.83.

\[ \begin{aligned} -\tfrac{7}{9} &\approx -0.778 \lt -0.75 \\ -\tfrac{1}{2} &= -0.5 \gt -0.75 \\ -\tfrac{2}{5} &= -0.4 \gt -0.75 \end{aligned} \]

So \(-\frac{7}{9}\) is to the left of a on the number line. 0, \(-\frac{1}{2}\) and \(-\frac{2}{5}\) all lie between \(-0.75\) and 0.83. Comparing \(\frac{7}{9}\) with \(\frac{3}{4}\) by cross-multiplication (\(28 \gt 27\)) gives the same result without decimals.

Answer: C

The largest 6-digit number has 5 distinct prime factors.

The largest 6-digit number is 999999.

\[ \begin{aligned} 999999 &= 999 \times 1001 \\ &= (3^{3} \times 37) \times (7 \times 11 \times 13) \end{aligned} \]

The distinct primes are 3, 7, 11, 13 and 37, which is five.

Missing the factor 37 (by stopping at \(999 = 27 \times 37\) without splitting it) is the usual way to get 4; counting powers of 3 as separate factors inflates the count.

Answer: C

The correct order is \(\frac{11}{13} \lt \frac{15}{17} \lt \frac{13}{14}\).

Each fraction is 1 minus a small part, so compare the parts left over: the fraction with the smaller gap from 1 is the larger.

\[ \begin{aligned} 1 - \tfrac{11}{13} &= \tfrac{2}{13} \approx 0.154 \\ 1 - \tfrac{15}{17} &= \tfrac{2}{17} \approx 0.118 \\ 1 - \tfrac{13}{14} &= \tfrac{1}{14} \approx 0.071 \end{aligned} \]

So \(\frac{11}{13}\) is the smallest and \(\frac{13}{14}\) the largest. In decimals they are about 0.846, 0.882 and 0.929.

The option \(\frac{11}{13} \lt \frac{13}{14} \lt \frac{15}{17}\) assumes that a fraction with bigger numbers is bigger, which is not true; the other two put \(\frac{11}{13}\) or \(\frac{13}{14}\) in the wrong place.

Answer: A

Statements a and c are correct, so the answer is (a) and (c).

A rational number is any number that can be written as p/q with p and q integers and q not zero. Every integer n is n/1, so the rational numbers include the integers (a). Every fraction (a/b with whole numbers, b not zero) is of this form, so all fractions are rational (c).

Statement b is false: 0 = 0/1 is a rational number. Options naming b are therefore wrong, and 'Only (c)' leaves out the true statement a.

Answer: B

The reciprocal lies between \(\frac{9}{50}\) and \(\frac{1}{5}\).

The other number is the product divided by the known number, and its reciprocal follows:

\[ \begin{aligned} \text{other} &= -\tfrac{40}{3} \div \left(-\tfrac{5}{2}\right) = \tfrac{40}{3} \times \tfrac{2}{5} = \tfrac{16}{3} \\ \text{reciprocal} &= \tfrac{3}{16} = 0.1875 \end{aligned} \]

Since 9/50 = 0.18 and 1/5 = 0.2, 0.1875 lies between them. 3/20 = 0.15 to 0.18 stops just short, and 1/5 to 1/4 and 1/4 to 1/3 are above it.

Answer: C

The value of 2p + q - r is 23.

Factorise 52272:

\[ \begin{aligned} 52272 &= 16 \times 3267 \\ &= 2^{4} \times 27 \times 121 \\ &= 11^{2} \times 3^{3} \times 2^{4} \end{aligned} \]

Matching with \(p^{2} \times q^{3} \times r^{4}\) gives p = 11, q = 3 and r = 2. Then

\[ 2p + q - r = 22 + 3 - 2 = 23 \]

21 comes from subtracting q and adding r the wrong way, and 22 is just 2p. 29 would need different primes in the factorisation.

Answer: C

The descending order is \(\frac{44}{49}, \frac{22}{25}, \frac{33}{38}, \frac{24}{29}\).

Each fraction is 1 minus a small fraction, so the fraction with the smallest gap from 1 is the largest:

\[ \begin{aligned} \tfrac{44}{49} &= 1 - \tfrac{5}{49} \approx 0.898 \\ \tfrac{22}{25} &= 1 - \tfrac{3}{25} = 0.880 \\ \tfrac{33}{38} &= 1 - \tfrac{5}{38} \approx 0.868 \\ \tfrac{24}{29} &= 1 - \tfrac{5}{29} \approx 0.828 \end{aligned} \]

The order 44/49, 33/38, 24/29, 22/25 wrongly puts 22/25 last because its numbers are small. The other two options run in ascending or mixed order.