E Maths: variations, algebra

lolaminx SG

i am back , again, and while revising, i came across a couple of questions and i need some clarification:

question 1

If 4 men can make 80 chairs in 12 days, how long will 24 men take to make 300 chairs?

My steps:

4 men --- 80 chairs -- 12 days

1 man -- 20 chairs -- 3 days (take 80 divide by 4 to get 20)

24 men -- 480 chairs -- 3 days (take 20 chairs x 24= 480)

24 men -- 480 chairs -- 3days

24 men - 300 chairs -- x days

24 men -- 300 chairs - 48 days (480 divide by 300 is 1.6, so i take 3 x 1.6= 4.8)

however, the answer given is 7.5 days.

question 2

5(x-y)^2 - 5x + 5y

= 5(x-y)(x+y) - 5(x+y)

= err then i kind of have no idea how to continue this. help >_<

question 3

Given that y varies directly as t and inversely as the square of x, and y=5 when t=2 and x=4... express y in terms of t and x.

y=kt y= 1/x^2

5=k(2) y= k/x^2

5/2=k 5=k/4^2

* so y= 5/2 t 5=k/16

80=k

* so y= 80/x^2

combining both together:

y= (80/x^2) x ((5/2) t) (and then multiply etc. to this answer:)

y+ 200/x^2 t

so yeah my question is: are my steps + answer for this question correct? i don't have the sample answers with me, so it wld be great if someone can help point out my mistake.

thanks everyone ^_^

skythewood SG

i am back , again, and while revising, i came across a couple of questions and i need some clarification:

question 1

If 4 men can make 80 chairs in 12 days, how long will 24 men take to make 300 chairs?

My steps:

4 men --- 80 chairs -- 12 days

1 man -- 20 chairs -- 3 days (take 80 divide by 4 to get 20)

24 men -- 480 chairs -- 3 days (take 20 chairs x 24= 480)

24 men -- 480 chairs -- 3days

24 men - 300 chairs -- x days

24 men -- 300 chairs - 48 days (480 divide by 300 is 1.6, so i take 3 x 1.6= 4.8)

however, the answer given is 7.5 days.

question 2

5(x-y)^2 - 5x + 5y

= 5(x-y)(x+y) - 5(x+y)

= err then i kind of have no idea how to continue this. help >_<

question 3

Given that y varies directly as t and inversely as the square of x, and y=5 when t=2 and x=4... express y in terms of t and x.

y=kt y= 1/x^2

5=k(2) y= k/x^2

5/2=k 5=k/4^2

* so y= 5/2 t 5=k/16

80=k

* so y= 80/x^2

combining both together:

y= (80/x^2) x ((5/2) t) (and then multiply etc. to this answer:)

y+ 200/x^2 t

so yeah my question is: are my steps + answer for this question correct? i don't have the sample answers with me, so it wld be great if someone can help point out my mistake.

thanks everyone ^_^

4 men make 80 chairs in 12 days

1 men make 20 chairs in 12 days

1 men make 5 chair in 3 days

24 men make 120 chairs in 3 days

24 men make 40 chairs in 1 day

24 men make 300 chairs in 7.5 days

skythewood SG

5(x-y)^2 - 5x + 5y

5(x-y)^2 + (-5x + 5y)

5(x-y)^2 - 5(x-y)

5 [ (x-y)^2 - (x-y)]

5(x-y) [ (x-y) -1]

Mikethm SG

Like you said it is a variation question.

The 2 variables are number of chairs and man/days.

Let chair be x and Mandays be y

More mandays = more chair. Thus direct variation.

y varies directly as x.

y = kx

When x = 80, y = 48

48= 80k

k = 0.6

y = 0.6x

When x = 300 y = 24a

24a = 0.6(300)

a = 7.5

iamapebble SG

4 men --- 80 chairs -- 12 days

NOTE: men inversely proportional to number of days,
men directly proportional to number of chairs.

So let's do it step by step (:

4 men -- 80 chairs -- 12 days
24 men -- 80 chairs -- 2 days
24 men -- 300 chairs -- 7.5 days (1 day -> 40 chairs)

Do they mean factorise? If yes:

5(x-y)^2 - 5x + 5y
= 5(x-y)(x-y) - 5x + 5y
= 5(x-y)(x-y) - 5(x-y)
= [5(x-y)] [(x-y) - 1]
= (5x - 5y)(x - y - 1)

NOTE: (x-y)^2 is not equals to (x-y)(x+y).

y = kt
5 = k(2)
k = 5/2
y = (5/2)t

y = k/(x^2)
5 = k/(4^2)
5 = k/(16)
k = 80
y = 80/x^2

y^2 = (5/2)t * 80/x^2
y = plus/minus square root [(5/2)t * 80/x^2]
y = plus/minus square root (200t/x^2)

You were careless as you times the two y together but you didnt do it to your y^2.

lolaminx SG

thanks everyone!

lolaminx SG

hi iampebble... so that means since (x-y)^2 equals to = (x-y)(x-y)... this applies to every algebra questions pertaining to such?

so that means it is only that for eg. (x^2 - 4) is (x+2)(x-2) right?

iamapebble SG

hi iampebble... so that means since (x-y)^2 equals to = (x-y)(x-y)... this applies to every algebra questions pertaining to such?

so that means it is only that for eg. (x^2 - 4) is (x+2)(x-2) right?

There are 3 general equations you have to take note of :

1) (a+b)^2 = (a+b)(a+b) = (a^2 + 2ab + b^2)
2) (a-b)^2 = (a-b)(a-b) = (a^2 - 2ab + b^2)
3) (a^2 - b^2) = (a+b)(a-b)

=) As long as you practise more you will get the feel of it and they will be a breeze to you in no time! And yes (x^2 - 4) is (x+2)(x-2), because x = a, and 4 = 2^2, which means b = 2. =)