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A company wants to distribute 14,500 Egyptian pounds among the top five sales representatives as a bonus.
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The bonus for the last-placed representative is 1,300 Egyptian pounds, and the difference in bonus is constant among the representatives.
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Find the bonus of the representative in the first place.
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We have five sales representatives ordered from first to fifth who are going to share this money.
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The bonus for the last-placed representative, thatβs the representative in fifth place, weβre told is 1,300 Egyptian pounds.
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Weβre also told that the difference in the bonus being paid is constant between the representatives, which means that these amounts form an arithmetic sequence with a common difference of π.
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We donβt know what this common difference is, but as the bonuses are decreasing, we know that its value will be negative.
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Weβve also been given the total amount of money that is going to be shared between these five people.
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This is the sum of the five terms in our arithmetic sequence.
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We know that there is a formula for calculating the sum of the first π terms in an arithmetic sequence.
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Itβs π sub π is equal to π over two multiplied by two π plus π minus one π, where π represents the first term in the sequence and π represents the common difference.
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We can therefore form an equation in π and π.
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Substituting 14,500 for the sum of the five terms and five for π, we have 14,500 is equal to five over two multiplied by two π plus four π.
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We can simplify this equation slightly by first canceling a factor of two on the right-hand side to give 14,500 equals five over one, or five, multiplied by π plus two π.
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And then we can divide both sides of the equation by five to give 2,900 is equal to π plus two π.
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Now we have an equation connecting π and π.
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But this isnβt enough information to enable us to work out π and π, as we have only one equation and two unknowns.
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The other information we were given is that the fifth term of the sequence is equal to 1,300.
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This tells us that if we take our first term π and we add the common difference of π four times, then we get 1,300.
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So we have the equation π plus four π equals 1,300.
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We now have a pair of linear equations in π and π which we can solve simultaneously.
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Subtracting the first equation from the second will eliminate the π terms and leave two π is equal to negative 1,600.
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We can then divide both sides of this equation by two to find that π is equal to negative 800.
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We have a negative value for π, as we expected, so thatβs reassuring.
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We can then substitute this value of π into either of our two equations, Iβve chosen equation one, to give an equation in π only.
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We have π minus 1,600 is equal to 2,900.
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Adding 1,600 to each side of this equation, we find that π is equal to 4,500.
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So weβve found the bonus paid to the representative in first place.
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To check our answer, letβs calculate the bonuses paid to the remaining representatives by subtracting 800 each time.
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This gives 3,700, 2,900, 2,100.
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And if we subtract 800 again, this does indeed give 1,300.
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If we then sum these five values together, it does indeed give 14,500, which is the correct total.
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So we can confirm that the bonus paid to the representative in first place is 4,500 Egyptian pounds.