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10. दुई अङ्कले बनेको एउटा सङ्ख्यामा एकको स्थानमा भएको अङ्क दशको स्थानमा भएको अङ्कको एक चौथाइ छ। अङ्कहरूको स्थान बदलिँदा बन्ने सङ्ख्या उक्त सङ्ख्याभन्दा 54 कम छ।

In a number of two digits, the digit in the units' place is one-fourth of the digit in the tens place. The number formed by reversing the digits is 54 less than the original number.

(a) दिइएको कथनलाई रेखीय समीकरणको रूपमा प्रस्तुत गर्नुहोस्।
Express the given conditions in the form of linear equations.
[tex]\[ [\text{Ans: } y = \frac{x}{4}, (x - y) = 6] \][/tex]

(b) उक्त सङ्ख्या पत्ता लगाउनुहोस्।
Find the number.
[tex]\[ [\text{Ans: } 82] \][/tex]

(c) विपरीत सङ्ख्या पनि पत्ता लगाउनुहोस्।
Find the reverse number too.
[tex]\[ [\text{Ans: } 28] \][/tex]



Answer :

GinnyAnswer
Let's solve the given problem step by step.

### Part (a): Express the Given Conditions in the Form of Linear Equations

1. Let [tex]\( x \)[/tex] be the digit in the tens place of the number.
2. Let [tex]\( y \)[/tex] be the digit in the units place of the number.

#### Condition 1:
The digit in the units' place ([tex]\( y \)[/tex]) is one-fourth of the digit in the tens place ([tex]\( x \)[/tex]):
[tex]\[ y = \frac{x}{4} \][/tex]

#### Condition 2:
When the digits are reversed, the new number formed is 54 less than the original number. If the original number is [tex]\( 10x + y \)[/tex], then the reversed number will be [tex]\( 10y + x \)[/tex]. Therefore:
[tex]\[ 10y + x = 10x + y - 54 \][/tex]

So, the equations we have are:
1. [tex]\( y = \frac{x}{4} \)[/tex]
2. [tex]\( 10y + x = 10x + y - 54 \)[/tex]

### Part (b): Find the Number

Now, let's solve these equations to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].

#### Using the first equation [tex]\( y = \frac{x}{4} \)[/tex]:
This implies [tex]\( y = \frac{x}{4} \)[/tex].

#### Substitute [tex]\( y = \frac{x}{4} \)[/tex] in the second equation:

[tex]\[ 10\left(\frac{x}{4}\right) + x = 10x + \left(\frac{x}{4}\right) - 54 \][/tex]

Simplify:

[tex]\[ \frac{10x}{4} + x = 10x + \frac{x}{4} - 54 \][/tex]
[tex]\[ \frac{10x + 4x}{4} = 10x + \frac{x}{4} - 54 \][/tex]
[tex]\[ \frac{14x}{4} = 10x + \frac{x}{4} - 54 \][/tex]
Multiply everything by [tex]\( 4 \)[/tex] to clear the denominators:

[tex]\[ 14x = 40x + x - 216 \][/tex]
[tex]\[ 14x = 41x - 216 \][/tex]
Rearrange to isolate [tex]\( x \)[/tex]:

[tex]\[ 14x - 41x = -216 \][/tex]
[tex]\[ -27x = -216 \][/tex]
[tex]\[ x = 8 \][/tex]

Now, substitute [tex]\( x = 8 \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:

[tex]\[ y = \frac{8}{4} \][/tex]
[tex]\[ y = 2 \][/tex]

Therefore, the original number is:
[tex]\[ 10x + y = 10(8) + 2 = 80 + 2 = 82 \][/tex]

### Part (c): Find the Reverse Number

The reverse number is formed by reversing the digits:

[tex]\[ 10y + x = 10(2) + 8 = 20 + 8 = 28 \][/tex]

### Final Answer

- The original number is 82.
- The reverse number is 28.

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