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4 years ago

Which rates are unit rates?

1 answer:

5 0

Unit rate is a rate that has a denominator of 1. In order to get the unit rate given the denominator is not 1, then unit rate is acquired by getting the equal ration with denominator 1. On the question, Letter A. 4 yd/carry, Letter B. 35 mi/1gal and Letter D. 3 cups per batch are unit rates.

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Find two positive numbers whose difference is 30 and whose product is 2584.

kvasek [131]

Answer:

$38$ and $68$ .

Step-by-step explanation:

Let $x$ be the smaller one of the two number.

$x$ must be a positive integer. The other number would be $(x + 30)$ .

The question states that the product of the two numbers is $2584$ . In other words:

$x\, (x + 30) = 2584$ .

Rearrange this equation and solve for $x$ :

$x^{2} + 30\, x - 2584 = 0$ .

The first root of this quadratic equation would be:

$\begin{aligned}x_{1} &= \frac{(-30) + \sqrt{30^{2} - 4 \times (-2584)}}{2} \\ &= \frac{(-30) + \sqrt{900 + 10336}}{2} \\ &= \frac{(-30) + \sqrt{11236}}{2} \\ &= \frac{(-30)}{2} + \sqrt{\frac{11236}{2^{2}}} \\ &= (-15) + \sqrt{2809} \\ &= (-15) + 53 \\ &= 38 \end{aligned}$ .

Similarly, the second root of this quadratic equation would be:

$\begin{aligned}x_{1} &= \frac{(-30) - \sqrt{30^{2} - 4 \times (-2584)}}{2} \\ &= (-15) - 53 \\ &= -68\end{aligned}$ .

Since the question requires that both numbers should be positive, $x > 0$ . Therefore, only $x = 38$ is valid.

Hence, the two numbers would be $38$ and $(38 + 30)$ , which is $68$ .

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