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Right choice is (a) \(\frac{139}{26}\)

The BEST I can explain: The numbers when PLACED on the number line, the number which is placed farthest from 0 is the greatest. Here the numbers are, \(\frac{139}{26}\) = 5.34

\(\frac{72}{21}\) = 3.42

\(\frac{99}{36}\) = 2.75

\(\frac{101}{20}\) = 5.05

So, we CONCLUDE that the greatest number is \(\frac{139}{26}\) and hence would be placed farthest from 0.

The CORRECT option is (d) \(\frac{12}{4}\) and \(\frac{13}{5}\)

To ELABORATE: Here we have to find the ordered pair of numbers and the order should be BIG following by small. This condition is only satisfied by one pair of numbers which is \(\frac{12}{4}\) and \(\frac{13}{5}\).

Correct option is (b) Integers

To explain: The given SET is of integers.

The sets of other numbers are,

Whole number -> [0, 1, 2, 3, 4, ……]

Natural number -> [1, 2, 3, 4, …….]

RATIONAL number -> [……, 0, ……..] This set consists of all numbers on the number LINE.

The CORRECT choice is (a) –\(\frac{139}{26}\)

Easiest explanation: The numbers when placed on the number line, the number which is placed nearest from 0 is the smallest. Here the numbers are, –\(\frac{139}{26}\) = -5.34

–\(\frac{72}{21}\) = -3.42

–\(\frac{99}{36}\) = -2.75

–\(\frac{101}{20}\) = -5.05

So, we conclude that the smallest number is –\(\frac{139}{26}\) and HENCE would be placed smallest from 0. Thing to remember: When the numbers are placed on the left side of the number line, the number farthest from the CENTER is the smallest number. This is due to the NEGATIVE sign.

Correct answer is (c) \(\frac{7}{2}\) and \(\frac{10}{2}\)

To elaborate: When we place the 4 given pairs and the number on the number LINE we observe that the number \(\frac{32}{9}\) LIES between the pair \(\frac{7}{2}\) and \(\frac{10}{2}\), while other options fail to capture the given number between them. Hence the correct pair is \(\frac{7}{2}\) and \(\frac{10}{2}\).

The correct option is (a) \(\frac{3}{1000}\)

The explanation is: The given fractional numbers represents \(\frac{3}{10}\) = 0.3 and \(\frac{3}{10000}\) = 0.00003

Our options are in fractions, converting it into decimal form we GET,

\(\frac{3}{1000}\) = 0.0003

\(\frac{3}{10000}\) = 0.00003

\(\frac{3}{100000}\) = 0.000003

\(\frac{3}{1000000}\) = 0.0000003.

The only option between 0.3 and 0.00003 is \(\frac{3}{1000}\) i.e. 0.0003.

The correct OPTION is (d) infinite

The BEST EXPLANATION: There can be an infinite number of rational numbers between any TWO consecutive NATURAL numbers. But there cannot be any integers between two consecutive natural numbers.

Correct answer is (B) False

To elaborate: The set of integers is […., -1, 0, 1, ….] there cannot be any integers between two integers.

Right OPTION is (a) \(\frac{36}{7}\)

The BEST I can explain: The only number which is not between 6 and 7 is \(\frac{36}{7}\)

\(\frac{36}{7}\) = 5.14

\(\frac{45}{7}\) = 6.42

\(\frac{46}{7}\) = 6.57

\(\frac{13}{2}\) = 6.5.

Correct answer is (a) \(\FRAC{1}{5}\)

Best explanation: The given fractional numbers represents \(\frac{1}{4}\) = 0.25 and \(\frac{1}{6}\) = 0.17

Our OPTIONS are \(\frac{1}{5}\) = 0.20

\(\frac{1}{6}\) = 0.17

\(\frac{1}{3}\) = 0.33

\(\frac{1}{2}\) = 0.50

The only option between 0.17 and 0.25 is \(\frac{1}{5}\) i.e. 0.20.

Correct answer is (d) smaller and smaller

Easy explanation: When the numbers -8 and 8 placed on the number line -8 would be on the left side of 0 and 8 would be on the right side of 0. Hence -8 is smaller than 8. Similarly, for -12 and -9, -12 is smaller than -9. This is because as we go TOWARDS left on a number line the magnitude of the number increases but due to its NEGATIVE sign it is smaller than the rest of the number to the right of that number.

The correct answer is (b) False

To EXPLAIN: The number line which consists of only integers, the set of integers is [……,-3, -2, -1, 0, 1, 2, 3, ……]Hence the number line won’t CONSIST of DECIMAL like 2.5. Hence the GIVEN statement is false.

Right ANSWER is (b) \(\frac{12}{11}\)

To elaborate: All the numbers given to us are positive and according to the sign convention they WOULD be placed on the right side of the 0. Here we have 4 numbers which are

\(\frac{5}{2}\) = 2.5

\(\frac{7}{6}\) = 1.67

\(\frac{5}{4}\) = 1.25

\(\frac{12}{11}\) = 1.09

Among these four decimal numbers the smallest is 1.09 which is decimal representation of \(\frac{12}{11}\), this will be nearest to the CENTER i.e. 0.

Right option is (d) \(\frac{-230}{300} \,and \,\frac{-46}{60}\)

Explanation: When the rational numbers in their reduced form give equal fractions those numbers are CALLED equivalent pair of rational numbers.

Here, we have 4 pairs of rational numbers in fractional form.

\(\frac{120}{45} \,and \,\frac{160}{6}\) when reduced we get 8/3 and 80/3

\(\frac{55}{6} \,and \,\frac{220}{12}\) when reduced we get 55/6 and 55/3

\(\frac{-8}{13} \,and \,\frac{16}{26}\)when reduced we get (-8)/13 and 8/13

\(\frac{-230}{300} \,and \,\frac{-46}{60}\) when reduced we get \(\frac{-23}{30} \,and \,\frac{-23}{30}\). Here we SEE that both the numbers are equal and hence this pair is equivalent pair.

Right choice is (c) the line that extends indefinitely only to the right side of 1

Explanation: The set of NATURAL numbers is [1, 2, 3, 4, 5…..] i.e. natural numbers start from 1 and extends to infinity on the positive side. The OPTION which STATES that ‘The line that extends indefinitely to the right, but from 0′ shows the number line for whole numbers.

Correct choice is (b) LEFT SIDE

The explanation: As we KNOW that on a number LINE the negative numbers are placed on the left side of the 0 and positive numbers on the right side. Here \(\frac{-16}{7}\) is a negative number and hence has to be placed on the left side. Other options are wrong as negative numbers cannot be place on right side or at the center.

Correct answer is (a) NEGATIVE -> 0 -> positive

The explanation is: When we refer to any number line, we OBSERVE that considering 0 as the center, the numbers to the left are negative and to the right are positive. The correct option WOULD be “negative -> 0 -> positive”.

Right answer is (b) -x

For EXPLANATION I WOULD say: The number OBTAINED by dividing a number by 1 is its reciprocal. All the NUMBERS have one and only one reciprocal number.

Right answer is (a) \(\FRAC{5}{2}\)

Explanation: The correct answer to the question is \(\frac{5}{2}\), since it is the only number that lies between 2 and 3. While the others don’t LIE in the given range.

Options \(\frac{6}{2}\) and \(\frac{6}{3}\) are not correct because the question states “between” and not “EQUAL to”.

Right ANSWER is (d) 9

Best explanation: \(\frac{15}{135}\) * x = 1

Therefore \(\frac{1}{9}\) * x = 1

Therefore x = 9

Multiplicative INVERSE of a PARTICULAR number, is the number which gives 1 on multiplication with the number. There is one and only one multiplicative inverse for all numbers.

Right ANSWER is (d) \(\FRAC{-5}{4}\)

To explain: Using the PROPERTIES of addition and SUBTRACTION we can solve this problem.

\(\frac{-15}{32} / x = \frac{12}{32}\)

Therefore x = \(\frac{-15}{32} / \frac{12}{32}\)

Therefore x = \(\frac{-15}{32} * \frac{32}{12}\)

Therefore x = \(\frac{-15}{12}\)

Therefore x = \(\frac{-5}{4}\).

Correct choice is (a) \(\frac{16}{21}\)

The best I can EXPLAIN: A rational number is said to be in it’s standard form when the numbers cannot be further reduced.

Here, we can SEE that the options other than the OPTION \(\frac{16}{21}\) can be reduced.

When \(\frac{16}{24}\) is reduced, we get \(\frac{2}{3}\)… Standard Form.

When \(\frac{21}{15}\) is reduced, we get \(\frac{7}{5}\)… Standard Form.

When \(\frac{-21}{56}\) is reduced, we get \(\frac{-3}{8}\)… Standard Form.

Right choice is (d) \(\frac{121}{16}\)

The EXPLANATION is: In order to simplify the given equation we have to equate the denominators.

We will do this by taking the LEAST common multiple.

Therefore \(\frac{11}{2} + \frac{11}{4} – \frac{11}{8} +\frac{11}{16} = \frac{88}{16} + \frac{44}{16} – \frac{22}{16} + \frac{11}{16}\)

= \(\frac{121}{16}\).

Right choice is (b) \(\frac{32}{15}\)

The best I can explain: Using the properties of ADDITION and subtraction we can solve this problem.

\(\frac{16}{15} – x = \frac{-16}{15}\)

THEREFORE \(\frac{16}{15} + \frac{16}{15}\) = x

Therefore x = \(\frac{32}{15}\)

Option \(\frac{-32}{15}\) cannot be the CORRECT answer, as the question itself ASKS to subtract the number in order to obtain \(\frac{-16}{15}\).

The CORRECT answer is (c) \(\frac{5}{3}\)

Explanation: Using the property of MULTIPLICATION, we can SOLVE the question.

\((\frac{15}{6}) * (\frac{2}{3})\)

= \((\frac{5}{2}) * (\frac{2}{3})\)

= 5/3.

Right choice is (a) 26

The best explanation: By USING the PROPERTIES of rational numbers like ADDITION and multiplication, we can solve the equation and get to the correct ANSWER.

(8+0) + (6 * 3)

= (8) + (18)

= 26.