Direct and inverse proportions

Question

1. Kate makes a donation to a charitable organisation on a monthly basis. Her monthly donation is directly proportional to the square of her monthly savings. If she saves $900 and $1200 in January and February respectively, her donation increases by $35 from January to February. Find the amount of money she donates to the charitable organisation in each of the two months.

2. 5 men are hired to complete a job. If one more man is hired, the job can be completed 8 days earlier. Assuming that all the men work at the same rate, how many more men should be hired so that the job can be completed 28 days earlier?

Answer 1

she saves 900 in january, and 1200 in february

her donation increases by 35 from january to february

her donation is directly proportional to the square of her savings

d = ks^2

d = k900^2 = 810000k

''for january

d = k1200^2 = 1440000k

for february

1440000k = 35 + 810000k

1440000k- 810000k = 35

630000k = 35

k = 35/630000 = 7/126000

in january d=(7/126000)900^2

= (7/126)810

= $45

in february d= (7/126000)1200^2

= $80

80-45=35

5 men are hired, by hiring a 6th man the job can be completed 8 days earlier

how many more men should be hired to complete the job 28 days earlier?

let x = the number of days it takes 5 men to complete the job

then x-8 = days for 6 men to complete the job

x-28 = days for y men to complete the job

each man works at the same rate

it takes 1 man alone to complete the job in 5x days

it takes 5 men to complete the job in 5x/5 = x days

it take 6 men to complete the job in x-8 days

z days for 1 man to complete the job

5 men do the job in z/5 days

6 men do the job in z/6 days

z/6 = z/5 -8

5z =6z -240

z = 240 days for 1 man to complete the job

5 men do the job in 240/5 = 48 days

6 men do the job in 240/6 = 40 days

to do the job in 48-28= 20 days

requires 240/y = 20

20y=240

y = 12 men

you need 12 men to do the job in 28 days less

so if you have 6, you need 6 more men