MATHEMATICAL OPERATIONS
Category : 7th Class
Learning Objectives:-
MATHEMATICAL OPERATIONS
This section deals with questions on simple mathematical operation. There are four fundamental operations, namely:
Addition i.e. \[+;\]Subtraction i.e. \[-;\]
Multiplication i.e, \[\times ;\]and Division i.e, \[\div \]
There are also statements such as Less than i.e \[<;\] greater than i.e, \[>;\] and equal to i.e, = not equal to i.e, \[\ne ;\] etc.
Such operations are represented by symbols, different from the usual ones. The questions involving these operations are coded using artificial symbols. The candidate has to make a substitution of real signs and solve the equation accordingly.
While solving a mathematical expression, we always proceed according to the rule B O D M A S. i.e, B for Brackets; O for' of (literarily multiplication),
D for division; M for multiplication, A for additions and S for subtraction.
DIFFERENT TYPES OF PROBLEMS
TYPE I : Problems-solving By Substitution
In this type, you provided substitutes for various mathematical symbols or numbers. Followed by question involving calculation of an expression or choosing the correct/ incorrect equations. The candidate is required to put the real signs or numerals in the given equation and then solve the questions as required.
Example 1:
(a) 153 (b) 216
(c) 248 (d) 251
Solution:
(a) Using the proper signs, we get
Given expression
\[=14\times 10+42-2-8=14\times 10+21-8\]
\[=140+21-8=161-8=153.\]
Example 2:
(a) \[15-5\div 5\times 20+10=6\]
(b) \[8\div 10-3+5\times 6=8\]
(c) \[6\times 2+3\div 12-3=15\]
(d) \[3\div 7-5\times 10+3=10\]
Solution:
(b) Using the proper signs, we get
Expression in (a) \[=15\times 5+5-20\div 10\]
\[=15\times 5+5-2=75+5-2=78\]
Expression in (b)
\[=8+10\times 3\div 5-6=8+10\times \frac{3}{5}-6=8+6-6=8.\]
Expression in (c)
\[=6-2\div 3+12\times 3=6-\frac{2}{3}+36=42-\frac{2}{3}=\frac{124}{3}\]
Expression in (d)
\[=3+7\times 5-10\div 3=3+7\times 5-\frac{10}{3}=\frac{104}{3}\]
\[\therefore \] Statement (b) is true
Example 3:
Which one of the four interchanges in signs and numbers would make the given equation correct?
\[3+5-2=4\]
(a) \[+\]and\[-,\]2 and 3 (b) \[+\]and\[-,\]2 and 5
(c) \[+\] and\[-,\]3 and 5 (d) None of these
(c) \[6\times 2+3\div 12-3=15\]
(d) \[3\div 7-5\times 10+3=10\]
Solution:
(c) By making the interchanges given in (a) we get the equations as \[2-5+3=4\] or \[0=4,\] which is false.
By making the interchanges given in (b) we gets the equations as \[3-2+5=4\] or \[6=4,\] which is false.
By making the interchanges given in (c) we get the equations as \[5-3+2=4,\] which is true.
So, the answer is (c)
Directions (Example 4 to 7): In each of the following examples which one of the four interchanges in signs and numbers would make the given equation correct?
Example 4:
\[6\times 4+2=16\]
(a) \[+\]and \[\times ,\]2 and 4
(b) \[+\]and \[\times ,\]2 and 6
(c) \[+\]and \[\times ,\]4 and 6
(d) None of these
Solution:
(c) On interchanging + and 4 and 6, we get the equation as
\[4+6\times 2=16\] or \[4+12=16\] or \[16=16,\] which is true
Examples 5:
\[\left( 3\div 4 \right)+2=2\]
(a) \[+\]and\[-,\]2 and 3
(b) \[+\]and\[-,\]2 and 4
(c) \[+\]and\[-,\]3 and 4
(d) No interchanges, 3 and 4
Solution:
(a) By interchanging + and - and 2 and 3, we get the equations as
\[\left( 2+4 \right)\div 3=2\]or \[6\div 3=2\] or \[2=2,\] which is true.
Example 6:
\[4\times 6-2=14\]
(a) \[\times \]to\[-,\]2 and 4
(b) \[-\]to\[-,\]2 and 6
(c) \[-\]to +, 2 and 6
(d) \[\times \]to\[+,\]4 and 6
Solution:
(c) On changing\[-\]to\[+\]and interchanging 2 and 6, we get the equation as
\[4\times 2+6=14\] or \[8+6=14\] or \[14=14\] which is true.
Example 7:
\[\left( 6\div 2 \right)\times 3=0\]
(a) \[-\]and\[\times ,\]2 and 3
(b) \[\times \]to\[-,\]2 and 6
(c) \[-\]and\[\times ,\]2 and 6
(d) \[\times \]to\[-,\]2 and 3
Solution:
(d) By changing \[\times \] to \[-\] and interchanging 2 and 3, we get the equations as
\[\left( 6\div 3 \right)-2=0\] or \[2-2=0\] or \[0=0,\] which is true.
MATHEMATICAL LOGIC
Considers statement" 5 is greater than 3.
Now consider which of the following statements are true and which are false.
"5 is not greater than 3" (False)
"5 is equal to3" (False)
"5 is less than 3" (False)
"5 is not equal to 3" (True)
"5 is not less than 3" (True)
In general, between any two numbers a and b, only one of the following relations can exist at a time
\[a>b\]
or \[a>b\]
or \[a=b\]
If \[a>b,\]then \[a\not{<}b\] and \[a\ne b\]
If \[a<b,\]then \[a\not{>}b\] and \[a\ne b\]
If \[a=b,\]then \[a\not{>}b\] and \[a\not{<}b\]
Directions for Example 8-9
Let the following symbols denote some relationship between numbers.
\[O=\]greater than \[\phi =\]not greater than
\[+=\]equal to \[A=\]not equal to
\[=\]less than \[\times =\]not less than
In the examples below, find the correct answer.
Examples 8:
If\[p\,\,q\,O\,r,\]it is possible that
(a) \[p\,\phi \,q\,r\] (b) \[p\,\phi \,q\times \,r\]
(c) \[p\,+\,q\times \,r\] (d) \[p\,\Delta \,q\,\phi \,r\]
Solution:
(b) \[\](less than \[\Rightarrow \Delta \] (Not equal to) or \[\phi \] (not greater than)
O (greater than) \[\Rightarrow \Delta \] (Not equal to) or \[\times \] (not less than)
For p, and q, option (a), (b), (d) are possible.
For q and r, options (b), (c) are possible.
Hence the answer is (b)
Example 9:
If \[p\,\Delta \,q\,O\,r,\]it is possible that
(a) \[p\times q\times r\]
(b) \[p\times q\,\,r\]
(c) \[p\,\,q\,\phi \,r\]
(d) \[p\,\phi \,q\,\phi \,r\]
Solution:
(a) \[\Delta =\]not equal to
Hence, \[\Delta \Rightarrow O\](greater than )
or \[\Delta \Rightarrow \] (Less than)
\[\Delta \Rightarrow \phi \] (not greater than)
or \[\Delta \Rightarrow x\](not less than)
Similarly, \[O\Rightarrow \Delta \] and \[O\Rightarrow x\]
Hence \[P\Delta \,q\Rightarrow p\,O\,q\]or\[p\,\,p\]or\[p\,\phi \,q\]or \[p\,\times \,q\]
and \[q\,O\,r\Rightarrow q\,\Delta \,\,r\]or\[q\times r\]
All four options are possible so far as p and q are concerned.
Between q and r, only the first is correct.
Example 10:
If\[+\]means\[\div ,\,\times \]means\[-,\,\times \]and\[-\]means\[+,\]then
\[8+6\times 4\div 3-4=\]
(a) \[-12\] (b) \[-20\text{/}3\]
(c) \[12\] (d) \[20\text{/}3\]
Solution:
(b) Given expression\[=8\div 6-4\times 3+4\]
\[=\frac{4}{3}-4\times 3+4=\frac{4}{3}-12+4=-\frac{20}{3}\]
Example 11:
If a means ‘Plus’ b means minus’c means' multiplied by’ and d means 'divide by’ then 18 c 14 a 6 b 16 d 4 = ?
(a) 63 (b) 254
(c) 288 (d) 1208
Solution:
(b) Given expression\[=18\times 14+6-16\div 4\]
\[=\frac{4}{3}-4\times 3+4=\frac{4}{3}-12+4=-\frac{20}{3}\]
Example 12:
If A stands for\[+,\]B stands for\[-,\] C stands for\[\times ,\] then what is the value of\[(10\,\text{C}4)A\,(4C4)\,\text{B}\,6?\]
(a) 60
(b) 56
(c) 50
(d) 20
Solution:
(c) Given expression \[=\left( 10\times 4 \right)+\left( 4\times 4 \right)-6=50\]
Example 13:
If A means \[-,\]B means \[\div ,\]C means \[+\] and D means\[\times ,\]the 15 B 3 C 24 A 12 D 2 = ?
(a) 34
(b) 2
(c) \[\frac{5}{9}\]
(d) 5
Solution:
(d) Given expression \[=15\div 3+24-12\times 2\]\[=5+24-24=5\]
TYPE II
Directions (Example 14-16): In each of the following questions, an equation becomes incorrect due to the interchange of two signs. One of the four alternatives under this rule specifies the interchange of sign in the equations which when made will make the equation correct Find the correct alternative.
Example 14:
\[16-8\div 4+5\times 2=8\]
(a) \[\div \]and\[\times \]
(b) \[-\]and\[+\]
(c) \[\div \]and\[\div \]
(d) \[-\]and\[\div \]
Solution:
(b) On interchanging\[-\]and\[\div \]we get:
Given expression\[=16\div 8-4+5\times 2=2-4+10=8\]
Example 15:
\[56\div 7\times 2+8-1=9\]
(a) \[\times \]and\[-\]
(b) \[\div \]and\[\times \]
(c) \[+\]and\[-\]
(d) \[+\]and\[\div \]
Solution:
(c) On interchanging - and +, we get
Given expression\[=56\div 7\times 2-8+1=16-8+1=9\]
Example 16:
\[121\div 11-3\times 13+2=22\]
(a) \[-\]and\[\times \]
(b) \[-\]and\[\div \]
(c) \[\div \]and\[-\]
(d) \[+\]and\[-\]
Solution:
(a) On interchanging\[-\]and\[\times \]we get
Given expression\[=121\div 11\times 3-13+2\]
\[=\frac{121}{11}\times 3-13+2=11\times 3-13+2=22\]
TYPE III
Direction (Example 17-19): In each of the following questions, three statements of numbers following same rules are given. Find the rule and according to it and find the value of the number?
Example 17:
If \[84\oplus 72=45;\]\[63\oplus 41=33,\] \[25\oplus 52=33,\]then\[94\oplus 82=?\]?
(a) 45
(b) 59
(c) 56
(d) 65
Solution:
(c) The rule is Difference of the digits of the number
\[84\oplus 72=(8-4)\,(7-2)=45\,\text{etc}\text{.}\]
Example 18:
If \[32\times 41=15;\]\[51\times 34=47;\]\[41\times 52=37,\] then\[87\times 53=\]?
(a) 68 (b) 64
(c) 85 (d) 18
Solution:
(d) The logic is \[32\times 41=(3-2)(4+1)=15;\]
\[51\times 34=(3+4)=47\]etc.
\[\therefore 87\times 53=(8-7)\,(5+3)=18\]
Example 19:
If \[5\times 9=144;\] \[7\times 8=151:4\times 6=102,\] then\[2\times 5=\]?
(a) 73 (b) 77
(c) 37 (d) 97
Solution:
(a) The rule is\[a\times b=\left( a+b \right)\left( b-a \right)\]
\[\therefore 2\times 5=(2\times 5)\,(5-2)=73\]
Example 20:
If \[\times \] stands for addition, \[<\] stands for subtraction, \[>\] stands for multiplication, + stands for division, 0 stands for greater than, = stands for less than and \[-\] stands for equal to then\[-\]
(a) \[3\times 2<4\,\text{O}\,6+3<2\] (b) \[3+2<4\,\text{O}\,6>3\times 2\]
(c) \[3>2<4-6\times 3\times 2\] (d) \[3\times 2\times 4=6+3<2\]
Solution:
(a) We may rewrite all the statements using the meaning of the symbols used. We therefore have
(a) \[3+2-4>+\,6\div 3-2\] (True)
We may check the other statements also to make sure that our answer is correct
(b) \[3\div 2-4>\,6\times 3+2\] (False)
(c) \[3\times 2-4=\,6+3+2\] (False)
(d) \[3+2+4<\,6\div 3-2\] (False)
Example 21:
If P denotes \[\div ;\] Q denotes \[\times ;\] R, denotes \[+;\] and S denotes \[-;\] then the value of 18 Q 12 P 4 R 5 S 6 when simplified gives
(a) 36 (b) 53
(c) 59 (d) 65
Solution:
(b) Using correct symbols, we have
Given expression \[=18\times 12\div 4+5-6\]
\[=18\times 3+5-6=54+5-6=53\]
Example 22:
If\[7*1=64;\]
\[3*9=144\]
What is the value of\[5*6\]?
(a) 22 (b) 55
(c) 66 (d) 121
Solution:
(d) Rule is \[a*b={{\left( a+b \right)}^{2}}\]\[\therefore 5*6={{\left( 5+6 \right)}^{2}}=121.\]
TYPE IV
Directions (Example 23-24): In each of the following questions, three statements of numbers following same rules are given. Find the rule and according find the value of the number
Example 23:
If \[2\times 1=81;\] \[3\times 2=278;\] \[2\times 5=8125,\] then \[1\times 3=\]
(a) 127 (b) 271
(c) 126 (d) 129
Solution:
(a) The rule is\[a\times b={{a}^{3}}{{b}^{3}}\]
\[2\times 1={{2}^{3}}{{1}^{3}}=81\]etc. So, \[1\times 3={{1}^{3}}{{3}^{3}}=127\]
Example 24:
If \[2\div 3=89;\] \[3\div 4=2716;\] \[4\div 3=649,\] then \[1\div 2=?\]
(a) 21 (b) 42
(c) 14 (d) 81
Solution:
(c) The rule is \[a\div b={{a}^{3}}{{b}^{3}}\] \[\therefore 1\div 2={{1}^{3}}{{2}^{2}}=14.\]
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