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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Step-by-step 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} = (300)/(x)\\\\`

If the unit price is x dollars per kg, and she spent $300 total, then she can buy
`(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} = (300)/(x-50)\\\\`

She's able to buy 0.5 kg more which means the
`(300)/(x)` increases to
`(300)/(x)+0.5` and this is set equal to the
`(300)/(x-50)` since both expressions describe the amount she can buy at this discount rate.

So we have this equation:

`(300)/(x-50) = (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.

`(300)/(x-50) = (300)/(x)+0.5\\\\x(x-50)*(300)/(x-50) = x(x-50)*\left((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 = (-b\pm√(b^2-4ac))/(2a)\\\\x = (-(-25)\pm√((-25)^2-4(0.5)(-15000)))/(2(0.5))\\\\x = (25\pm√(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.