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zzz [600]
3 years ago
5

3. At a bargain store, Tanya bought 4 items that each cost the same amount. Tony bought 5 items that each cost the same amount,

but each was $1.25 less than the items that Tanya bought. Both Tanya and Tony paid the same amount of money. What was the individual cost of each person's items? (a) Write an equation. Let x represent the cost of one of Tanya's items. (b) Solve the equation. Show your work. (c) Check your solution. Show your work. (d) State the solution in complete sentences
Mathematics
1 answer:
FinnZ [79.3K]3 years ago
6 0

We'll do it in cents and convert when we have to.

Tanya paid x per item, a total of 4x.

Tony paid x-125 per item (in cents) for a total of 5(x-125)

4x = 5(x-125)

4x = 5x - 625

x = 625 = $6.25  

Check:

4(625) = 2500

5(625-125)=2500, same, checks

Answer: Tanya paid $6.25 per item, Tony $5.00 per item


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How many permutations of three items can be selected from a group of six? Use the letters A, B, C, D, E, and F to identify the i
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In total, 120 permutations of three items can be selected from a group of six distinct elements.

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\begin{aligned}\rm B-D-F \\ \rm B-F-D \\ \rm D-B-F \\ \rm D-F-B \\ \rm F-B-D \\ \rm F-D-B\end{aligned}.

Step-by-step explanation:

The formula \displaystyle P(n,\, r) = \frac{n!}{(n - r)!} = n \, (n - 1) \cdots (n - r + 1) gives the number of ways to select and order r items from a group of n distinct elements.

To select and order three items from a group six distinct elements, let n = 6 and r = 3. Apply the formula:

\begin{aligned} P(6,\, 3) &= \frac{6!}{(6 - 3)!} = \frac{6!}{3!} \\ &= \frac{6 \times 5 \times 4 \times 3\times 2 \times 1}{3 \times 2 \times 1} \\ &= 6 \times 5 \times 4 = 120 \end{aligned}.

In other words, there are 120 unique ways to select and order three items (select a permutation of three items) from a group of six distinct elements.

Consider: what's the number of ways to order three distinct items? That's the same as asking: how many ways are there to select and order three items from a group of three distinct elements? Let n =3 and r = 3. Apply the formula for permutation:

\begin{aligned} P(3,\, 3) &= \frac{3!}{(3 - 3)!} = \frac{3!}{0!} && \left(\text{$0! = 1$ by convention.}\right) \\ &= 3! = 3 \times 2\times 1 \\ &= 6\end{aligned}.

To find the permutations, start by selecting one element as the first of the list. A tree diagram might be helpful. Refer to the attachment for an example.

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