Answer

Solution

UPSC Prelims

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CSAT 2023

Q9 2023, Number System

If ABC and DEF are both 3-digit numbers such that A, B, C, D, E and F are distinct non-zero digits such that ABC + DEF = 1111, then what is the value of A + B + C + D + E + F ?

यदि ABC और DEF दोनों ही 3-अंकों की संख्याएँ हैं, इस प्रकार कि A, B, C, D, E और F भिन्न शून्येतर अंक हैं, इस प्रकार कि ABC + DEF = 1111, तो A + B + C + D + E + F का मान क्या है ?

Explanation Please wait..

ABC + DEF = 1111, with A, B, C, D, E, and F being different non-zero digits.

To achieve a sum of 1111,

C + F = 11, for example, 2 + 9 = 11.

B + E = 10, like 3 + 7 = 10.

A + D = 10, for instance, 4 + 6 = 10.

Verifying the solution by adding 432 + 679 = 1111.

So, A + B + C + D + E + F = 4 + 3 + 2 + 6 + 7 + 9 = 31

Therefore, the correct option is (d).

Explanation Approach 2:

Step by step calculation:

ABC and DEF are 3-digit numbers, and both are distinct non-zero digits. It means

We are given that ABC + DEF = 1111. Substituting the expressions for ABC and DEF, we get:

Rearranging the terms, we get:

Since the sum of three-digit numbers can't exceed 999 + 999 = 1998, we know that 𝐴+𝐷 must be either 1 or 0.

If 𝐴+𝐷=1 => 𝐵+𝐸=11, and C+F=11, and

If 𝐴+𝐷=0, then B+E=12 and 𝐶+F=11.

Let's consider the case 𝐴+𝐷=1. => We have two possibilities for B+E=11: (5, 6) or (6, 5).

Let's try both cases:

If 𝐴+D=1, B+E=11, and C+F=11 with (5, 6), then ABC + DEF becomes 156 + 955 = 1111. This satisfies the condition.

If 𝐴+𝐷=1, B+E=11, and C+F=11 with (6, 5), then ABC + DEF becomes 165 + 946 = 1111. This also satisfies the condition.

Now, let's calculate the sum of digits 𝐴+𝐵+𝐶+𝐷+𝐸+F for both cases:

𝐴+𝐵+𝐶+𝐷+𝐸+𝐹 = 1+5+6+9+5+5 = 31

𝐴+𝐵+𝐶+𝐷+𝐸+𝐹 = 1+6+5+9+4+6 = 31

Both cases gives a sum of 31 for the digits.

So, the correct answer is (d) 31.

<ul><li> 
ABC
+ DEF = 1111, with A, B, C, D, E, and F being different non-zero digits.</li></ul>

<ul><li> 
To
achieve a sum of 1111,</li></ul>

<ul><li> 
C
+ F = 11, for example, 2 + 9 = 11.</li></ul>

<ul><li> 
A
+ D = 10, for instance, 4 + 6 = 10.</li></ul>

<ul><li> 
Verifying
the solution by adding 432 + 679 = 1111.</li></ul>

<ul><li> 
Therefore,
the correct option is (d).</li></ul>

<h5>Explanation
Approach 2:</h5>

Step
by step calculation:

ABC and
DEF are 3-digit numbers, and both are distinct non-zero digits. It means

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</v:imagedata></v:shape>

<v:shape id="_x0000_i1025" type="#_x0000_t75">
 <v:imagedata src="file:///C:\Users\GAURAV~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.png" o:title="" chromakey="white">
</v:imagedata></v:shape>

We are
given that ABC + DEF = 1111. Substituting the expressions for ABC and DEF, we
get:

<v:shape id="_x0000_i1025" type="#_x0000_t75">
 <v:imagedata src="file:///C:\Users\GAURAV~1\AppData\Local\Temp\msohtmlclip1\01\clip_image003.png" o:title="" chromakey="white">
</v:imagedata></v:shape>

Rearranging
the terms, we get:

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</v:imagedata></v:shape>

Since the
sum of three-digit numbers can't exceed 999 + 999 = 1998, we know that 𝐴+𝐷 must be either 1 or 0.

Let's
consider the case 𝐴+𝐷=1.  => We have two
possibilities for B+E=11: (5, 6) or (6, 5).

Let's
try both cases:

<ul><li> 
If 𝐴+D=1,
B+E=11, and C+F=11 with (5, 6), then ABC + DEF
becomes 156 + 955 = 1111. This satisfies the condition.</li></ul>

<ul><li> 
If 𝐴+𝐷=1, B+E=11,
and C+F=11 with (6, 5), then ABC + DEF becomes 165 + 946 = 1111.
This also satisfies the condition.</li></ul>

Now, let's
calculate the sum of digits 𝐴+𝐵+𝐶+𝐷+𝐸+F for both cases:

Both cases
gives a sum of 31 for the digits.

So, the
correct answer is (d) 31.

CSAT Year Wise (Temporary Link, This will be removed and no PDFs shall be downloaded here. Please download CSAT subject wise files) / सी सैट

Q9 2023, Number System

If ABC and DEF are both 3-digit numbers such that A, B, C, D, E and F are distinct non-zero digits such that ABC + DEF = 1111, then what is the value of A + B + C + D + E + F ?

Explanation Please wait..

Explanation Approach 2:

व्याख्या Please wait..

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John Doe

CSAT Year Wise (Temporary Link, This will be removed and no PDFs shall be downloaded here. Please download CSAT subject wise files) / सी सैट

Q9 2023, Number System

If ABC and DEF are both 3-digit numbers such that A, B, C, D, E and F are distinct non-zero digits such that ABC + DEF = 1111, then what is the value of A + B + C + D + E + F ?

Explanation Please wait..

Explanation Approach 2:

व्याख्या Please wait..

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