Question

Mrs. Ford spent $300 to buy some premium tea leaves. If
the price of the tea leaves had been reduced by $50 per kg,
she could have bought 0.5 kg more. Find the original price
per kg of the tea leaves. with steps please

Answer 1

Answer:  $200 per kg

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Explanation:

$x = \text{original price per kg}\\\\\text{amount spent} = (\text{unit price})*(\text{amount purchased})\\\\\text{amount purchased} = \frac{\text{amount spent}}{\text{unit price}}\\\\\text{amount purchased} = \frac{300}{x}$

If the unit price is x dollars per kg, and she spent $300 total, then she can buy $\frac{300}{x}$ kg of tea leaves. This is before the discount.

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Now let's consider the discount.

$x-50 = \text{unit price after discount of \ 50 per kg}\\\\\text{amount purchased} = \frac{\text{amount spent}}{\text{unit price}}\\\\\text{amount purchased} = \frac{300}{x-50}$

She's able to buy 0.5 kg more which means the $\frac{300}{x}$ increases to $\frac{300}{x}+0.5$ and this is set equal to the $\frac{300}{x-50}$ since both expressions describe the amount she can buy at this discount rate.

So we have this equation:

$\frac{300}{x-50} = \frac{300}{x}+0.5$

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Let's do a bit of algebra to get things into standard form.

Multiply both sides by x(x-50) to clear out the denominators.

$\frac{300}{x-50} = \frac{300}{x}+0.5\\\\x(x-50)*\frac{300}{x-50} = x(x-50)*\left(\frac{300}{x}+0.5\right)\\\\300x = 300(x-50) + 0.5x(x-50)\\\\300x = 300x - 15000 + 0.5x^2 - 25x\\\\0.5x^2 - 25x - 15000 = 0$

From here, use the quadratic formula with

a = 0.5, b = -25, c = -15000

$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-(-25)\pm\sqrt{(-25)^2-4(0.5)(-15000)}}{2(0.5)}\\\\x = \frac{25\pm\sqrt{30625}}{1}\\\\x = 25\pm175\\\\x = 25+175 \ \text{ or } \ x = 25-175\\\\x = 200 \ \text{ or } \ x = -150$

It doesn't make sense to have a negative unit price, so we cross x = -150 off the list. Only x = 200 is the practical answer.

The unit price is $200 per kg.

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Checking the answer:

Before the discount, she's able to buy 300/x = 300/200 = 1.5 kg of tea leaves.

After the discount, the unit price is x-50 = 200-50 = 150 dollars per kg and she's able to buy 300/150 = 2 kg of tea leaves. Note the jump from 1.5 kg to 2 kg is exactly an increase of 0.5 kg. This confirms we have the correct original unit price.