Setting Up the Age Equations
This problem involves finding the sum of the ages of two people, A and B. We can represent their ages with variables. Let's denote the age of A as a years and the age of B as b years.
The problem gives us two pieces of information, which we can translate into mathematical equations:
- "The sum of the age of A and 5 times the age of B is 36 years." This translates to the equation: $a + 5b = 36 \quad (\text{Equation 1})$
- "When 3 times the age of A is added to 7 times the age of B, the result is 62 years." This translates to the equation: $3a + 7b = 62 \quad (\text{Equation 2})$
Our goal is to find the sum of their ages, which is a + b.
Solving the System of Equations
We have a system of two linear equations with two variables:
- \( a + 5b = 36 \)
- \( 3a + 7b = 62 \)
We can solve this system using the elimination method. The idea is to eliminate one of the variables (either a or b) by manipulating the equations.
Let's eliminate variable a. To do this, we can multiply Equation 1 by 3 so that the coefficient of a in both equations becomes the same.
Multiply Equation 1 by 3:
$3 \times (a + 5b) = 3 \times 36$ $3a + 15b = 108 \quad (\text{Equation 3})$Now we have:
| Equation 3: | \( 3a + 15b = 108 \) |
| Equation 2: | \( 3a + 7b = 62 \) |
Subtract Equation 2 from Equation 3 to eliminate a:
$(3a + 15b) - (3a + 7b) = 108 - 62$ $3a + 15b - 3a - 7b = 46$ $8b = 46$Now, solve for b:
$b = \frac{46}{8}$ $b = \frac{23}{4}$Now that we have the value of b, we can substitute it back into either Equation 1 or Equation 2 to find the value of a. Let's use Equation 1:
$a + 5b = 36$ $a + 5 \left( \frac{23}{4} \right) = 36$ $a + \frac{115}{4} = 36$To find a, subtract \(\frac{115}{4}\) from both sides:
$a = 36 - \frac{115}{4}$To subtract, find a common denominator, which is 4:
$a = \frac{36 \times 4}{4} - \frac{115}{4}$ $a = \frac{144}{4} - \frac{115}{4}$ $a = \frac{144 - 115}{4}$ $a = \frac{29}{4}$Calculating the Sum of Ages
We have found the values for a and b:
- Age of A, \( a = \frac{29}{4} \) years
- Age of B, \( b = \frac{23}{4} \) years
The question asks for the sum of the ages of A and B, which is \( a + b \).
$a + b = \frac{29}{4} + \frac{23}{4}$Since the denominators are the same, we can add the numerators:
$a + b = \frac{29 + 23}{4}$ $a + b = \frac{52}{4}$Simplify the fraction:
$a + b = 13$Therefore, the sum of the ages of A and B is 13 years.
Important Questions from Problem on Ages (Notes)
- एक व्यक्ति की वर्तमान आयु उसकी माँ की आयु का $2/9$ वाँ भाग है । 10 वर्ष बाद, वह अपनी माँ की आयु का $4/11$ वाँ भाग हो जाएगा । 15 वर्ष बाद माँ की आयु कितनी होगी?
- एक आदमी अपने बेटे से 24 साल बड़ा है । दो साल बाद, उसकी उम्र उसके बेटे की उम्र से दोगुनी हो जाएगी । उसके बेटे की वर्तमान आयु है
- एक पिता ने अपने बेटे से कहा, "मैं तुम्हारे जन्म के समय तुम्हारी वर्तमान उम्र के बराबर था।" यदि पिता की उम्र अब 38 वर्ष है, तो पाँच वर्ष पहले बेटे की उम्र थी
- समीर और आनंद की वर्तमान आयु क्रमशः $5:4$ के अनुपात में है । तीन साल बाद, उनकी आयु का अनुपात क्रमशः $11:9$ हो जाएगा । आनंद की वर्तमान आयु वर्षों में क्या है ?
- कमला अपने छोटे भाई से दो गुना बड़ी है। यदि उनके बीच के आयु में अंतर 15 वर्ष है, तो उसके छोटे भाई की आयु क्या होगी ?
