Question

PLZ HELP ME ASAP PLZ PLZ PLZ!!!!! I NEED LOTS OF HELP!!! I USED 40 POINTS

Sarah, John, and David each choose a different favorite positive fraction. Each fraction is in simplest form. Johns's fraction is a reciprocal of Sarah's. David mutiples his fraction by Sarah's and finds a product of 12/35. Then, he multiplies his fraction by John's number and get a product of 15/7. What is each person's favorite faction? Please write ur answer in full sentences.

Plz help me!!
Thanks!!

Answer 1

first, some defintions
reciprocals are like a/b and b/a or 5/1 and 1/5

remember,
`((a)/(b))/((c)/(d))=((a)/(b))((d)/(c))=(ad)/(bc)`

john's fraction is reciprocal of sarah's
john=a/b
sarah=b/a

david multiplies his by sarah and gets 12/35
(b/a)(david)=12/35

multplies his number by john and gets 12/35
(a/b)(david)=15/7

hmmm

(b/a)(david)=12/35 and
(a/b)(david)=15/7

if we divide them then we do

`(((b)/(a))(david))/(((a)/(b))(david))=((12)/(35))/((15)/(7))=`

`((b)/(a))/((a)/(b))=((12)/(35))/((15)/(7))=`

`((b)/(a))((b)/(a))=((12)/(35))((7)/(15))=`

`(b^2)/(a^2)=(84)/(525)=`

`(b^2)/(a^2)=(4)/(25)`
square root both sides

`(b)/(a)=(2)/(5)`

therefor
b=2 and a=5

john's number is a/b=5/2
sarah's number is 2/5

david, hmm

sarah times david=12/35
2/5 times david=12/35
times both sides by 5/2
david=60/70
david=6/7

check other

john times david=15/7
5/2 times 6/7=15/7
30/14=15/7
15/7=15/7
true





sarah's favorite number is 2/5
john's favorite number is 5/2
david's favorite number is 6/7