MHB Solving Word Problem - Linear equations

[paulmdrdo1]

I'm solving a plenty list of exercises about application of linear equation. Currently I have solved 97% of them correctly. But there are some items I really need help with.here are they,

1. A nugget of gold and quartz weighs 100 grams. Gold weighs 19.3 g/cm3 (grams per cubic centimeter), quartz weighs 2.6g/cm3, and the nugget weighs 6.4g/cm3. Find the weight of gold in the nugget.

2. A cold cream sample weighing 8.41 grams lost 5.83 grams of moisture on heating to 110°C. The residue on extracting with water and drying lost 1.27 grams of water-soluble glycerin. The balance was oil. Calculate the percentage composition of this cream.

I'll post the other two problems in another thread!

hoping for your quick response and help. thanks!

 

[Evgeny.Makarov]

I'm solving a plenty list of exercises about application of linear equation. Currently I have solved 97% of them correctly.

Good job!

1. A nugget of gold and quartz weighs 100 grams. Gold weighs 19.3 g/cm3 (grams per cubic centimeter), quartz weighs 2.6g/cm3, and the nugget weighs 6.4g/cm3. Find the weight of gold in the nugget.

The terminology in these problems leaves a lot to be desired. A quantity like 6.4g/cm3 is density, and a body cannot "weigh" such quantity. In any case, it is wrong to use the same word "weigh" with quantities of different dimensions, such as 100 g and 19.3 g/cm3. Further, weight is a force and is measured in Newtons; in contrast, mass is measured in grams. So the problem should ask to find the mass of gold.

Anyway, let $x$ be the mass of gold in the nugget. Express the following quanities through $x$.

1. The mass of quartz in the nugget.
2. The volume of gold.
3. The volume of quartz.
4. The total volume of the nugget.
5. The denisty (mass / volume) of the nugget.

Equate the last quantity to 6.4g/cm3 to get an equation in $x$.

 

[paulmdrdo1]

$100-x =$ mass of quartz in the nugget

$\frac{x}{19.3}=$ volume of gold

$\frac{100-x}{2.6} =$ volume of quartz

$\frac{x}{19.3}+\frac{100-x}{2.6}=6.4$

x= 96 grams mass of gold in the nugget. --- is this correct?

 

[Evgeny.Makarov]

$\frac{x}{19.3}+\frac{100-x}{2.6}=6.4$

Here the left-hand side is volume, but the right-hand side is density. The correct equation is
\[
100/\left(\frac{x}{19.3}+\frac{100-x}{2.6}\right)=6.4
\]
and the answer is $x=68.6$.

 

[paulmdrdo1]

yes. I forgot to change the density of nugget into volume. but this was what I had i mind,

$\frac{x}{19.3}+\frac{100-x}{2.6}=\frac{100}{6.4}$

x=68.6 grams

how about the second problem. please help me to get started. thanks!~

 

[Evgeny.Makarov]

2. A cold cream sample weighing 8.41 grams lost 5.83 grams of moisture on heating to 110°C. The residue on extracting with water and drying lost 1.27 grams of water-soluble glycerin. The balance was oil. Calculate the percentage composition of this cream.

I am not sure I understand the problem, in particular, the phrase "on extracting with water and drying". The best I understand it is as follows: The cream sample weighing 8.41 grams contains 5.83 grams of water, 1.27 grams of glycerin, and the rest is oil. You need to find the percentage of each of the three components.

 

[paulmdrdo1]

How do I start?

 

[Evgeny.Makarov]

Start by finding the mass of oil in the sample.

 

[paulmdrdo1]

my solution,

5.83+1.27+x=8.41

x = 1.31 grams of oil.

5.83/8.41 = 69.3 % water

1.27/8.41 = 15.1% glycerin

1.31/8.41 = 15.6% oil

please check! thanks!

 

[Evgeny.Makarov]

Yes, that's correct.