Number System
- Arrange the following fractions into ascending order.
4 ,− 2 ,− 7 , 5 3 9 8 12
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Decimal equivalents of fractions :
4 = 1.3 3 −2 = −0.2 9 −7 = −0.875 8 5 = 0.42 12
Correct Option: A
Decimal equivalents of fractions :
4 = 1.3 3 −2 = −0.2 9 −7 = −0.875 8 5 = 0.42 12
∴ – 0.875 < – 0.2 < 0.42 < 1.3i.e., −7 < − 2 < 5 < 4 8 9 12 3
- A certain number when successively divided by 3, 5 and 8 leaves remainder 1, 2, 3
respectively. Find the remainders when the same number is divided by reversing the divisors.
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This problem can be solved by determining true or complete remainder and dividing it by reversing the order of divisors.
True remainder = d1d2r3 + d1r2 + r1
Here, d1 = 3, d2 = 5, r1 = 1, r2= 2, r3 = 3
∴ True remainder = 3 × 5 × 3 + 3 × 2 + 1= 45 + 6 + 1 = 52
Now, we divide 52 by 8, 5 and 3
Hence, the remainders are 4, 1 and 1.Correct Option: B
This problem can be solved by determining true or complete remainder and dividing it by reversing the order of divisors.
True remainder = d1d2r3 + d1r2 + r1
Here, d1 = 3, d2 = 5, r1 = 1, r2= 2, r3 = 3
∴ True remainder = 3 × 5 × 3 + 3 × 2 + 1= 45 + 6 + 1 = 52
Now, we divide 52 by 8, 5 and 3
Hence, the remainders are 4, 1 and 1.
- What will be the unit’s digit in the product of (2467)153 × (341)72 ?
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The unit’s digit in (2467)153 = The unit’s digit in (7)153 = The
unit’s digit in (7)1 = 7
and the unit’s digit in (341)72 = 1
Because for any index to 1, the value of unit’s digit will be 1.
∴ The unit’s digit in the product of (2467)153 × (341)72 = 7 × 1 = 7Correct Option: C
The unit’s digit in (2467)153 = The unit’s digit in (7)153 = The
unit’s digit in (7)1 = 7
and the unit’s digit in (341)72 = 1
Because for any index to 1, the value of unit’s digit will be 1.
∴ The unit’s digit in the product of (2467)153 × (341)72 = 7 × 1 = 7
- The sum of all the 2-digit numbers is :
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The two-digit numbers are : 10, 11, 12, ..... 97, 98, 99.
As we know that,Sum of N natural Numbers 1 + 2 + 3 ........ + n = n ( n + 1) 2
As we know from the given question,
Required sum = 10 + 11 + 12 + .......... + 99
We can write as below,
∴ Required sum = ( 1 + 2 + 3 +......+ 99 ) – ( 1 + 2 + ..... + 9 )Correct Option: D
The two-digit numbers are : 10, 11, 12, ..... 97, 98, 99.
As we know that,Sum of N natural Numbers 1 + 2 + 3 ........ + n = n ( n + 1) 2
As we know from the given question,
Required sum = 10 + 11 + 12 + .......... + 99
We can write as below,
∴ Required sum = ( 1 + 2 + 3 +......+ 99 ) – ( 1 + 2 + ..... + 9 )∴ Required sum = 99 ( 99 + 1 ) − 9 ( 9 + 1 ) 2 2
∴ Required sum = 4950 – 45 = 4905
- What is the sum of two consecutive even numbers, the difference of whose square is 84?
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Let us assume that two consecutive even numbers are P and P + 2.
According to given question,
(P + 2)2 – P2 = 84
⇒ P2 + 4P + 4 – P2 = 84
2nd method
Let us assume that two consecutive even numbers are P and P + 2.
According to given question,
(P + 2)2 – P2 = 84
Use the formula a2 – b2 = ( a + b )( a - b)Correct Option: C
Let us assume that two consecutive even numbers are P and P + 2.
According to given question,
(P + 2)2 – P2 = 84
⇒ P2 + 4P + 4 – P2 = 84
⇒ 4P = 84 – 4 = 80⇒ P = 80 = 20 4
⇒ P + 2 = 20 + 2 = 22
∴ The required sum = 20 + 22 = 42
2nd method
Let us assume that two consecutive even numbers are P and P + 2.
According to given question,
(P + 2)2 – P2 = 84
Use the formula a2 – b2 = ( a + b )( a - b)
⇒ ( P + 2 + P )( P + 2 - P ) = 84
⇒ ( P + 2 + P )( 2 ) = 84
⇒ ( P + 2 + P ) = 84/2
⇒ ( P + 2 + P ) = 42
