KCSE exam question help

[Kiki]

Howdy! I'm trying to help out my niece with her maths and well, I'm just really bad. I've run into your website and hope you can help us out!
Here are two problems thus far:
1. The surface area S of a sphere is directly proportional to the square of its radius r. The volume V of the sphere is directly proportional to the cube of its radius r. Find the percentage increase in V when S increases by 25%.
2. Mary bought five more fifty-cent stamps than two shilling stamps. She also bought one-quarter as many five-shilling stamps as two-shilling stamps. She spent a total of sh 47.50. Determine how much more money she spent a total of sh 47.50. Determine how much more money she spent on the five-shilling stamps than on the five-shilling stamps than on the fifty-cent stamps.

Thank you guys so much!

 

[Deleted member 4993]

Howdy! I'm trying to help out my niece with her maths and well, I'm just really bad. I've run into your website and hope you can help us out!
Here are two problems thus far:
1. The surface area S of a sphere is directly proportional to the square of its radius r. The volume V of the sphere is directly proportional to the cube of its radius r. Find the percentage increase in V when S increases by 25%.
2. Mary bought five more fifty-cent stamps than two shilling stamps. She also bought one-quarter as many five-shilling stamps as two-shilling stamps. She spent a total of sh 47.50. Determine how much more money she spent a total of sh 47.50. Determine how much more money she spent on the five-shilling stamps than on the five-shilling stamps than on the fifty-cent stamps.

Thank you guys so much!

To help effectively, we need to see her work - so that we may know where to begin to help.

For problem 1 - start with V = k * r[sup:1iwxip8c]3[/sup:1iwxip8c]

for problem 2 - how many cents make a shilling?

 

[Kiki]

Thank you so much for the speedy reply!

Q. 1 -- Trying to work it out with your info.

Q. 2 -- There are 100 cents in 1 shilling.

 

[Deleted member 4993]

Howdy! I'm trying to help out my niece with her maths and well, I'm just really bad. I've run into your website and hope you can help us out!
Here are two problems thus far:
1. The surface area S of a sphere is directly proportional to the square of its radius r. The volume V of the sphere is directly proportional to the cube of its radius r. Find the percentage increase in V when S increases by 25%.
2. Mary bought five more fifty-cent stamps than two shilling stamps. She also bought one-quarter as many five-shilling stamps (??)as two-shilling stamps. She spent a total of sh 47.50. Determine how much more money she spent(?? compared to what?) a total of sh 47.50. Determine how much more money she spent on the five-shilling stamps(??) than on the five-shilling stamps than on the fifty-cent stamps.

Thank you guys so much!

Can you please check #2 problem for accuracy? It does not quite make sense!! Please review your post prior to submitting.

 

[Kiki]

Hi Subhotosh, I've re-checked the question and I typed it as is. It is possible the textbook has an error! This is a Kenyan book and author, unfortunately. Please don't stress yourself. I am truly appreciative.
On another note, can you help us out with the other question?
Thanks,
Kiki.

 

[mmm4444bot]

I typed it as is.

It is possible the textbook has an error

Hi Kiki. It is obvious to me that the book contains errors because some of their English is clearly wrong.



Mary bought five more fifty-cent stamps than two-shilling stamps.

This sentence makes sense.



She also bought one-quarter as many five-shilling stamps as two-shilling stamps.

This sentence makes sense.



She spent a total of sh 47.50.

Determine how much more money she spent a total of sh 47.50.

The first of these two sentences makes sense, but the second sentence is simply a repeat of the first, with the incomplete statement "Determine how much more money" added at the beginning. Hence, the English in the second sentence is clearly wrong.



Determine how much more money she spent on the five-shilling stamps than on the five-shilling stamps than on the fifty-cent stamps.

This sentence begins with a repeat of the same phrase added to the beginning of the previous sentence. This phrase is still incomplete; so, it makes no sense in either of its two locations!

Also, the second sentence above includes the nonsensical phrase "on the five-shilling stamps than on the five-shilling stamps".

Perhaps you did not recognize these textbook errors because English is not your primary language?

It helps to know that the exercise is Kenyan, too. On my first read, I thought that the exercise referred to British shillings and US cents.

Kenya divides its shilling into 100 units called cents, just as the US divides its dollar into 100 units called cents. Therfore, in Kenya, 50 cents is half a shilling.

And, to be clear, a proper notation for Kenyan shillings is "K Sh".

Hence, Mary paid a total of K Sh 47.50 for the three groups of stamps.



I'm going to GUESS that the instruction in exercise #2 is supposed to be, "Determine how much more Mary spent on her five-shilling stamps than she did on her 50-cent stamps".

So, we need to determine first how many stamps of each kind were purchased. After we know how many of each kind, we can easily determine the requested difference, yes?

The information given about the number of 50-cent stamps purchased AND about the number of five-shilling stamps purchased is stated in terms of the number of two-shilling stamps purchased. This means that we can express all three of these numbers in terms of the same symbol.



We begin by assigning a symbol to represent each of the three unknown numbers.

Let T = the number of two-shilling stamps purchased

Let F = the number of five-shilling stamps purchased

Let C = the number of 50-cent stamps purchased

Now, we are told that C is five more than T. Therefore, we can write:

C = T + 5

We are told that F is one-fourth of T. Therefore, we can write:

F = T/4

We just expressed each of the numbers C and F in terms of the number T. This is an important step because we want to form an equation that contains only ONE symbol.



Next, the amount of money that Mary spent on each group of stamps is calculated by multiplying the price-per-stamp times the number of stamps.

Let's do that for all three groups of stamps.

For the five-shilling stamps, the amount that Mary spent is expressed by (5)(T/4).

She spent (2)(T) on the group of two shilling stamps.

She spent (0.5)(T + 5) on fifty-cent stamps.



These three amounts (expressions) must add up to 47.50.

Can you write the equation, and solve it for T ?

If so, do you understand how knowing the value of T can lead you to answer the question (that is, the question as I've guessed it to be) ?

Otherwise, if I wrote anything that you do not understand, please ask specific questions.

Cheers!

 

[mmm4444bot]

Yikes!

I worked through exercise #1, and it took me nearly 45 minutes to correctly simplify the algebra involved (using my method).

Note: I began my work with the surface equation instead of the volume equation, so maybe Subhotosh is thinking of a different approach.


S = k[subs4vpa5v]s[/subs4vpa5v] * r^2

V = k[subs4vpa5v]v[/subs4vpa5v] * r^3

where symbol k[subs4vpa5v]s[/subs4vpa5v] represents the constant of proportionality in the area relationship and

symbol k[subs4vpa5v]v[/subs4vpa5v] represents the constant of proportionality in the volume relationship.



My strategy was to first solve the surface equation for r in terms of S and k[subs4vpa5v]s[/subs4vpa5v].

Substitute my resulting expression for the radius into the volume equation, obtaining an expression for the original volume in terms of S, k[subs4vpa5v]s[/subs4vpa5v], and k[subs4vpa5v]v[/subs4vpa5v].

Next, increase the number S by 25%, to get a different expression for the increased volume.

(Kiki, do you know how to express the number S increased by 25% ?)



I finally calculated the percent change using the formula:

(increased volume - original volume)/(original volume)

All symbols S, k[subs4vpa5v]s[/subs4vpa5v] and k[subs4vpa5v]v[/subs4vpa5v] cancel out.



My answer is: a sphere's volume increases by 39.75%, when its surface area increases by 25%.

I would like to see somebody confirm this.



:idea: Here's a thought. The values of the proportionality constants k[subs4vpa5v]s[/subs4vpa5v] and k[subs4vpa5v]v[/subs4vpa5v] are both well-known.

You might want to first try this exercise using the famous values, instead:

S = 4 * Pi * r^2

V = 4/3 * Pi * r^3

This will make the algebraic simplifications easier. (Should I have written "simplifyings" instead, Soroban? Heh, heh, heh)

You could then repeat the same exercise symbolically; that is, by using symbols to represent the constants 4Pi and 4Pi/3, for extra practice and double-checking.

What do you think?