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1. We bought 2 pens and 6 pencils, the ratio of the amount we spent on pencils to the amount we spent on pens was 7:4. Later we went back and bought 4 pencils and 3 pens and spent 1 dollar less. How much money did we spend that time?

 

2. An ammonia and water mixture fills a 5 gallon container. 80% of the mixture is ammonia but some of the mixture will be drained and replaced with pure water. If a 5 gallon mixture of fifty percent ammonia is desired, how many quarts of the mixture need to be drained before the water is added? (A quart is one-fourth of a gallon.) Express your answer as a decimal to the nearest tenth.

 

3. Not as important ~ I have difficulty understanding this question: 
There was a flat containing boxes of apples having a total weight of 100 kg. An analysis showed that the apples were 99% moisture, by weight. After two days in the sun, a second analysis showed that the moisture content of the apples was only 98%, by weight. What was the total weight of the apples after 2 days, in kg?

 

Thanks!! smiley

 
Guest Sep 11, 2018
 #1
avatar+88775 
+1

1. We bought 2 pens and 6 pencils, the ratio of the amount we spent on pencils to the amount we spent on pens was 7:4. Later we went back and bought 4 pencils and 3 pens and spent 1 dollar less. How much money did we spend that time?

 

Let the cost of a pen  = x  and the cost of  a pencil  = y

So we have that  

6y / 2x  = 7 / 4

3y / x  = 7/4

3y  = (7/4)x

y = (7/12)x

 

And we know that

2x + 6y   = A        where A  is in dollars

2x + 6(7/12)x  = A

2x + (42/12)x  = A

2x + 3.5x  = A

5.5x   = A  = (11/2)x    (1)

 

And we also know that

3x + 4y  =  A  - 1

3x + 4(7/12)x  = A  - 1

3x + (28/12)x  = A  - 1

3x + (7/3)x   = A  - 1

(16/3)x  = A  - 1      sub( 1)  in for  A

(16/3)x = (11/2)x- 1    subtract (11/2)x from both sides

-1/6x  = -1

x  = 6 (dollars)  = cost of a pen

And y = (7/12)(6)  = 42/12 =  3.5  dollars = cost of a pencil

 

So...the first time we spent

2(6) + 6(3.5) = $33

 

And the second time we spent

3(6) + 4(3.5)  = $32

 

 

cool cool cool

 
CPhill  Sep 11, 2018
 #2
avatar+88775 
+1

2. An ammonia and water mixture fills a 5 gallon container. 80% of the mixture is ammonia but some of the mixture will be drained and replaced with pure water. If a 5 gallon mixture of fifty percent ammonia is desired, how many quarts of the mixture need to be drained before the water is added? (A quart is one-fourth of a gallon.) Express your answer as a decimal to the nearest tenth.

 

5 gallons  = 20 qts  ....and 80% ammonia   = 20% water

 

Let x be the number of quarts of mixture that we need to drain

So...when this is done we have    [20 -x] qts of 20% water

[20 (.20) - x(.20)]  =

[20 - x] .20

 

To this, we obviously need to add back  "x" quarts of  pure water  to get 20 qts of  50% water [ = 50% ammonia ]

 

So we have

 

[20 - x ] .20  +  x (1)  = 20(.50)  simplify

 

4  - .2x + x  = 10

.8x + 4  = 10    subtract  4 from each side

.8x + 6     dibide both sides by .8

6 / [8/10]  = x

60/ 8  = x

x  = 7.5

 

So...we need to drain 7.5 qts  of the 20%water soluiton...and then add back this many qts of pure water to get a 20 qt solution of 50% water

 

 

cool cool cool

 
CPhill  Sep 11, 2018
edited by CPhill  Sep 11, 2018
 #3
avatar+27 
+1

I think 98 kg for the 3rd one 

 

 

 

 

 

wink

 
SVS2652  Sep 11, 2018
 #4
avatar
0

Here is my naive approach to this:
Weight of apples =99% moisture + 1%"F", for "fibre". Originally
After 2 days =98% moisture + 1%F =99 kg of their original weight.
The loss of 1 kg was solely due to moisture loss in the sun!!.

 
Guest Sep 11, 2018
 #5
avatar+1163 
+1

There was a flat containing boxes of apples having a total weight of 100 kg. An analysis showed that the apples were 99% moisture, by weight. After two days in the sun, a second analysis showed that the moisture content of the apples was only 98%, by weight. What was the total weight of the apples after 2 days, in kg?

----------

My statistics professor often posed questions like this to reinforce the importance of paying attention to how a question presents data and correctly interpreting data that results from analysis.

 

Solution:

 

\(\text {The first analysis shows the apples are 99% water. The weight of the water is then}\\ \left(0.99\cdot 100\right) = 99 ~ kg\\ \text {Let x be the weight of the water lost after exposure to the sun. }\\ \left(0.99 \cdot 100-0.98(100-x)\right)=x\\ \begin{aligned}99-(98-0.98x)&=x\\99-98+0.98x&=x\\1+0.98x&=x\end{aligned}\\ \begin{aligned}1+0.98x-0.98x&=x-0.98x\\1&=0.02x\\x&=50\\\end{aligned}\\ \text { }\\ 100-x=100-50=50 ~ kg \leftarrow \text { The total weight of the apples after 2 days}\\\)

 

The correct solution appears paradoxical. It’s not, but it is counter-intuitive.

This question is known as the Potato Paradox. We genetically enhanced Chimps refer to it as the Banana Paradox.smiley

 

 

GA

 
GingerAle  Sep 12, 2018
 #6
avatar+33 
+1

I don't understand the step \(0.99\cdot 100-0.98(100-x)=x\), why do you do this? Can you explain more? Thanks.

 
HelpPls  Sep 17, 2018
 #7
avatar+93289 
+1

I'll take a look :)

 

3. Not as important ~ I have difficulty understanding this question: 
There was a flat containing boxes of apples having a total weight of 100 kg. An analysis showed that the apples were 99% moisture, by weight. After two days in the sun, a second analysis showed that the moisture content of the apples was only 98%, by weight. What was the total weight of the apples after 2 days, in kg?

 

\(\text {The first analysis shows the apples are 99% water. The weight of the water is then}\\ \left(0.99\cdot 100\right) = 99 ~ kg\\ \text {Let x be the weight of the water lost after exposure to the sun. }\\ \left(0.99 \cdot 100-0.98(100-x)\right)=x\\ \)

 

You want this last step explained. I shall try.  

 

there are 100kg of apples and originally 99% of this is water so there is  0.99*100=99kg of water originally in the apples.

Over time the water dries out. All the weight that the apples lose is because of the reduced water content. 

After 2 days the water contant is only 98% of the weight. All the rest of the apple is still there.

 

Now in that 2 days GingerAle  has let x represents the unknown weight LOSS of the apples.

So after 2 days the apples will weigh  (100-x) kg

98% of this will be water so the new weight will be  98% of (100-x) = 0.98(100-x)

Weight after 2 days = 0.98(100-x)

 

So the original weight was (0.99*100)kg and the new weight is  0.98(100-x) and the weight loss is x

so

it follows that

0.99*100  -   0.98(100-x)   = x

 

I hope that helps.

 
Melody  Sep 17, 2018
 #8
avatar+93289 
+1

That final answer of weighing only 50kg after 2 days is so weird.

It does seem to be right though.

 

Originally 

weight = 100kg   water (99%) = 99kg

after 2 days

weight = 50kg    water (98%) = 49 kg

 
Melody  Sep 17, 2018

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