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lozanna [386]
2 years ago
12

The sparky valley barbershop quartet made appearances in Quakertown and Doleystown. The hotel charge before taxes in Doleystown

was $500 higher than Quakertown. The tax in Quakertown is 3% and Doleystown is 6%. The total hotel tax from the two places is $480. How much did each hotel charge the Quartet before taxes?
Mathematics
1 answer:
joja [24]2 years ago
5 0

Answer:

The hotel at Qakertown charged $5000 before tax while the hotel at Doleystown charged $5,500 before tax

Step-by-step explanation:

In this question, we have a musical band that visited two different towns which hotel charge different amounts at different tax rate. We are now given the total amount in taxes and now asked to calculate what is charged in each of the towns.

Let’s proceed!

Since we do not know the amount charged in each of the towns, let the amount charged at the hotel at Quakertown be $x and the amount charged at the hotel at Doleytown be $y.

Doley charged $500 more then what is charged at Quaker.

Mathematically;

y = 500 + x •••••••••••••••(I)

Let’s now work with the taxes;

Tax rate at Doley is 6% or say 0.06; the amount of tax paid at Doley is thus 0.06y

Tax rate at Quaker is 3% or say 0.03; the amount of tax paid at Quaker is thus 0.03x

Mathematically;

0.06y + 0.03x= 480 •••••••••(ii)

We can substitute i into ii

0.06(500 + x ) + 0.03x = 480

30+ 0.06x + 0.03x = 480

30 + 0.09x = 480

0.09x = 480 - 30

0.09x = 450

x = 450/0.09

x = $5,000

y = 500 + x

y = 500 + $5000 = $5,500

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Could the inverse of a non-function be a function? Explain or give an example.
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Answer:

The inverse of a non-function mapping is not necessarily a function.

For example, the inverse of the non-function mapping \lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\! is the same as itself (and thus isn't a function, either.)

Step-by-step explanation:

A mapping is a set of pairs of the form (a,\, b). The first entry of each pair is the value of the input. The second entry of the pair would be the value of the output.  

A mapping is a function if and only if for each possible input value x, at most one of the distinct pairs includes x\! as the value of first entry.

For example, the mapping \lbrace (0,\, 0),\, (1,\, 0) \rbrace is a function. However, the mapping \lbrace (0,\, 0),\, (1,\, 0),\, (1,\, 1) \rbrace isn't a function since more than one of the distinct pairs in this mapping include 1 as the value of the first entry.

The inverse of a mapping is obtained by interchanging the two entries of each of the pairs. For example, the inverse of the mapping \lbrace (a_{1},\, b_{1}),\, (a_{2},\, b_{2})\rbrace is the mapping \lbrace (b_{1},\, a_{1}),\, (b_{2},\, a_{2})\rbrace.

Consider mapping \lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\!. This mapping isn't a function since the input value 0 is the first entry of more than one of the pairs.

Invert \lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\! as follows:

  • (0,\, 0) becomes (0,\, 0).
  • (0,\, 1) becomes (1,\, 0).
  • (1,\, 0) becomes (0,\, 1).
  • (1,\, 1) becomes (1,\, 1).

In other words, the inverse of the mapping \lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\! would be \lbrace (0,\, 0),\, (1,\, 0),\, (0,\, 1),\, (1,\, 1) \rbrace\!, which is the same as the original mapping. (Mappings are sets. There is no order between entries within a mapping.)

Thus, \lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\! is an example of a non-function mapping that is still not a function.

More generally, the inverse of non-trivial ellipses (a class of continuous non-function \mathbb{R} \to \mathbb{R} mappings, including circles) are also non-function mappings.

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