Determine the combination of $GH \notin 20$ and GH $\not \subset 50$ notes to make GH $\not \subset 8,200$.
Answer (2)
Define variables: Let x be the number of GH 20 notes and y be the number of GH 50 notes.
Set up the equation: 20 x + 50 y = 8200 , which simplifies to 2 x + 5 y = 820 .
Solve for x in terms of y : x = 410 − 2 5 y . Since x must be an integer, y must be even, so let y = 2 k .
Substitute y = 2 k into the equation for x : x = 410 − 5 k . One possible solution is when k = 1 , giving x = 405 and y = 2 . Thus, a combination of 405 GH 20 notes and 2 GH 50 notes makes GH 8200.
Explanation
Define variables We are asked to find a combination of GH 20 and GH 50 notes that add up to GH 8200. Let's define our variables: Let x be the number of GH 20 notes and y be the number of GH 50 notes.
Set up the equation We can set up the equation: 20 x + 50 y = 8200 . To simplify, we divide the entire equation by 10: 2 x + 5 y = 820 .
Solve for x in terms of y Now, let's solve for x in terms of y : 2 x = 820 − 5 y , which gives us x = 410 − 2 5 y . Since x must be an integer, y must be an even number. Let y = 2 k , where k is an integer.
Substitute y = 2k Substitute y = 2 k into the equation for x : x = 410 − 2 5 ( 2 k ) = 410 − 5 k . Since x and y must be non-negative, we have x ≥ 0 and y ≥ 0 . This gives us 410 − 5 k ≥ 0 and 2 k ≥ 0 .
Solve the inequalities Solve the inequalities: 5 k ≤ 410 , so k ≤ 82 . Also, k ≥ 0 . Thus, 0 ≤ k ≤ 82 . The possible values for k are integers from 0 to 82.
Find a combination Let's pick a value for k . For example, let k = 1 . Then, y = 2 k = 2 ( 1 ) = 2 , and x = 410 − 5 k = 410 − 5 ( 1 ) = 405 . So, one combination is 405 GH 20 notes and 2 GH 50 notes.
Check the answer Let's check our answer: 20 ( 405 ) + 50 ( 2 ) = 8100 + 100 = 8200 . This is correct.
Another solution Another possible solution: Let k = 0 . Then y = 2 ( 0 ) = 0 and x = 410 − 5 ( 0 ) = 410 . So, 410 GH 20 notes and 0 GH 50 notes also make GH 8200.
One more solution Let's provide one more solution. Let k = 82 . Then y = 2 ( 82 ) = 164 and x = 410 − 5 ( 82 ) = 410 − 410 = 0 . So, 0 GH 20 notes and 164 GH 50 notes also make GH 8200.
Final Answer We can choose any value of k between 0 and 82 (inclusive) to find a valid combination of GH 20 and GH 50 notes. Let's choose the solution where k = 1 , which gives us 405 GH 20 notes and 2 GH 50 notes.
Conclusion A combination of 405 GH 20 notes and 2 GH 50 notes makes GH 8200.
Examples
Imagine you are a treasurer for a school event and need to prepare the float for the cashiers. You know you need a total of GH 8200 in GH 20 and GH 50 notes. By using the equation 20 x + 50 y = 8200 , where x is the number of GH 20 notes and y is the number of GH 50 notes, you can determine the different combinations of notes you can use. For instance, you could have 405 GH 20 notes and 2 GH 50 notes, or 410 GH 20 notes and no GH 50 notes. This ensures you have the correct denominations to provide change efficiently during the event.
To achieve GH 8200 using GH 20 and GH 50 notes, you can have combinations like 405 GH 20 notes and 2 GH 50 notes, or 410 GH 20 notes with no GH 50 notes, or 0 GH 20 notes and 164 GH 50 notes. Each of these combinations correctly totals GH 8200. Multiple combinations exist depending on chosen values for the variables.
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