SOLUTION: After a lady is seated in a​ restaurant, she realizes that she only has ​$23.00. If she must pay​ 7% sales tax and wishes to leave a 15​% tip on the total bill​ (meal plu

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Question 1196355: After a lady is seated in a​ restaurant, she realizes that she only has ​$23.00. If she must pay​ 7% sales tax and wishes to leave a 15​% tip on the total bill​ (meal plus​ tax), what is the maximum price of the lunch she can​ order?
The total price of lunch she can afford is ​$____
Found 2 solutions by math_tutor2020, MathTherapy:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Edit: @MathTherapy is not correct. The tip is calculated AFTER the tax is computed (due to the phrasing "meal plus tax"). If it said "meal excluding tax", then the equation would be P+0.15P+0.07P = 23 which solves to P = 18.85
The equation 1.15P + 1.07P = 23 is close, but not quite correct for the "meal excluding tax" scenario.

The answer is still $18.69

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x = price before tax and before tip

The multiplier 1.07 represents an increase of 7% since
100% + 7% = 1 + 0.07 = 1.07
For example, if the lunch was $10, then a 7% increase leads to 1.07*10 = 10.70 dollars

If the price before tax is x, then 1.07x is the price after tax
This is before the tip.

Multiply by 1.15 to increase by 15% and it adds on the tip (this only works if the tip includes tax)
1.15*1.07x = 1.2305x
This represents a total net increase of 23.05%
Unfortunately it's not as simple as saying 15% + 7% = 22%, but the 22% is a bit close to 23.05%

The expression 1.2305x represents the final cost after tax and after the tip. Set this equal to the $23 she has and solve for x.

1.2305x = 23
x = 23/(1.2305)
x = 18.6915887850468 approximately
x = 18.69
Divide both sides by 1.2305 to isolate x (step 2)

The highest price she can afford, before tax and before tip, is $18.69

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Checking the answer:

If the meal costs $18.69 before tax and before tip, then
7% of 18.69 = 0.07*18.69 = 1.3083 = 1.31 is the tax
The cost of the meal after tax, but before tip, is 18.69+1.31 = 20

Or you could use the shortcut mentioned earlier
1.07*18.69 = 19.9983 = 20
I'm rounding each value to the nearest penny, i.e. nearest hundredth decimal place.

Now let's increase that by 15% to find the tip
15% of 20 = 0.15*20 = 3
The tip is $3 to get a total bill of 20+3 = 23

Shortcut:
1.15*20 = 23

Also note this much quicker shortcut
1.2305*18.69 = 22.998045 = 23
Since we calculated earlier the tax and tip represent a net increase of 23.05%
So there are a few ways to verify the answer.
Whichever route you took, you should get to the target of 23 and it confirms the answer is correct.

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Answer: $18.69
Answer by MathTherapy(10555)   (Show Source): You can put this solution on YOUR website!

After a lady is seated in a​ restaurant, she realizes that she only has ​$23.00. If she must pay​ 7% sales tax and wishes to leave a 15​% tip on the total bill​ (meal plus​ tax), what is the maximum price of the lunch she can​ order?
The total price of lunch she can afford is ​$____
Be aware that tips are subject to INCOME TAX but NOT sales tax.

Let maximum price be P
Then a 15% tip will yield 1.15P
In addition, the maximum meal price, plus sales tax will be 1.07P
Since she has $23 to spend, we then get: 1.15P + 1.07P = 23
1.22P = 23
Maximum meal price, or 
=====
Check
=====
                    Meal cost:  $18.85
     Tip on $18.85: .15(18.85) =  2.83 (rounded)
SALES tax on $18.85: .07(18.85) = 1.32 (rounded)
                                $23.00
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