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!! **

Guest Sep 11, 2018

#1**+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

CPhill
Sep 11, 2018

#2**+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

CPhill
Sep 11, 2018

#4**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**+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. Itâ€™s not, but it is counter-intuitive.

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

GA

GingerAle
Sep 12, 2018

#6**+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**+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