Get the most out of your questions with the extensive resources available on IDNLearn.com. Find the answers you need quickly and accurately with help from our knowledgeable and experienced experts.
Sagot :
To find the least possible number of adult men in KZN, we start by establishing the relationship between the number of adult men and adult women based on the given information:
1. According to the problem, [tex]\(\frac{2}{3}\)[/tex] of the adult men are married to [tex]\(\frac{3}{5}\)[/tex] of the adult women.
2. Let [tex]\(m\)[/tex] represent the number of adult men and [tex]\(w\)[/tex] represent the number of adult women.
3. The relationship translates to:
[tex]\[ \frac{2}{3} \cdot m = \frac{3}{5} \cdot w \][/tex]
4. Simplifying this equation, we get:
[tex]\[ 2m = \frac{9}{5}w \][/tex]
Multiplying both sides by 5 to clear the fraction:
[tex]\[ 10m = 9w \][/tex]
5. Solving for [tex]\(w\)[/tex] in terms of [tex]\(m\)[/tex], we have:
[tex]\[ w = \frac{10}{9}m \][/tex]
We need the number of women, [tex]\(w\)[/tex], to be an integer. Starting with at least 300 men, we check if there is an integer value for [tex]\(w\)[/tex] corresponding to an integer [tex]\(m \geq 300\)[/tex]:
1. If we set [tex]\(m = 300\)[/tex]:
[tex]\[ w = \frac{10}{9} \cdot 300 = \frac{3000}{9} \approx 333.33 \quad \text{(not an integer)} \][/tex]
Since [tex]\(333.33\)[/tex] is not an integer, [tex]\(m = 300\)[/tex] is not a solution. We increment [tex]\(m\)[/tex] until we find the smallest [tex]\(m\)[/tex] making [tex]\(w\)[/tex] an integer:
2. Increasing [tex]\(m\)[/tex] step by step:
3. If we set [tex]\(m = 301\)[/tex]:
[tex]\[ w = \frac{10}{9} \cdot 301 = \frac{3010}{9} \approx 334.44 \quad \text{(not an integer)} \][/tex]
4. If we set [tex]\(m = 302\)[/tex]:
[tex]\[ w = \frac{10}{9} \cdot 302 = \frac{3020}{9} \approx 335.56 \quad \text{(not an integer)} \][/tex]
5. Continuing this process:
6. We find that when [tex]\(m = 306\)[/tex]:
[tex]\[ w = \frac{10}{9} \cdot 306 = \frac{3060}{9} = 340 \quad \text{(an integer)} \][/tex]
Thus, the least possible number of adult men in the province, ensuring that the number of adult women is an integer, is:
[tex]\[ \boxed{306} \][/tex]
1. According to the problem, [tex]\(\frac{2}{3}\)[/tex] of the adult men are married to [tex]\(\frac{3}{5}\)[/tex] of the adult women.
2. Let [tex]\(m\)[/tex] represent the number of adult men and [tex]\(w\)[/tex] represent the number of adult women.
3. The relationship translates to:
[tex]\[ \frac{2}{3} \cdot m = \frac{3}{5} \cdot w \][/tex]
4. Simplifying this equation, we get:
[tex]\[ 2m = \frac{9}{5}w \][/tex]
Multiplying both sides by 5 to clear the fraction:
[tex]\[ 10m = 9w \][/tex]
5. Solving for [tex]\(w\)[/tex] in terms of [tex]\(m\)[/tex], we have:
[tex]\[ w = \frac{10}{9}m \][/tex]
We need the number of women, [tex]\(w\)[/tex], to be an integer. Starting with at least 300 men, we check if there is an integer value for [tex]\(w\)[/tex] corresponding to an integer [tex]\(m \geq 300\)[/tex]:
1. If we set [tex]\(m = 300\)[/tex]:
[tex]\[ w = \frac{10}{9} \cdot 300 = \frac{3000}{9} \approx 333.33 \quad \text{(not an integer)} \][/tex]
Since [tex]\(333.33\)[/tex] is not an integer, [tex]\(m = 300\)[/tex] is not a solution. We increment [tex]\(m\)[/tex] until we find the smallest [tex]\(m\)[/tex] making [tex]\(w\)[/tex] an integer:
2. Increasing [tex]\(m\)[/tex] step by step:
3. If we set [tex]\(m = 301\)[/tex]:
[tex]\[ w = \frac{10}{9} \cdot 301 = \frac{3010}{9} \approx 334.44 \quad \text{(not an integer)} \][/tex]
4. If we set [tex]\(m = 302\)[/tex]:
[tex]\[ w = \frac{10}{9} \cdot 302 = \frac{3020}{9} \approx 335.56 \quad \text{(not an integer)} \][/tex]
5. Continuing this process:
6. We find that when [tex]\(m = 306\)[/tex]:
[tex]\[ w = \frac{10}{9} \cdot 306 = \frac{3060}{9} = 340 \quad \text{(an integer)} \][/tex]
Thus, the least possible number of adult men in the province, ensuring that the number of adult women is an integer, is:
[tex]\[ \boxed{306} \][/tex]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.