Hi all I have been trying to work this out for 3 days and seem to be missing something and get a different answer no matter what do.
Please help
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That's Latex ($\LaTeX$) the typesetting language used to render mathematical notation on the site. It's not mathematical notation.
Your error was in trying to remove the denominator from the fraction. You did it wrong, but also it was the wrong thing to do. You should first look to group like terms. In this case you have terms in $m$ and constant terms, so you need to combine all the terms in $m$ into a single term and all the constant terms into a single term.
Remember that $\frac{3m+10}{4}=\frac{3m}{4} + \frac{10}{4} = \frac34 m + \frac52$.
There are 4 unknown values, the number of men, m, the number of women, w, the number of boys, b, and the number of girls, g. We are given 4 pieces of information so can form 4 equations to solve for those 4 values.
"Three times as many women as men are on the tour": w= 3m.
"There are 10 less girls than the number of women": g= w- 10.
"The number of boys is a quarter of the number of girls": b= g/4.
"There are 65 people booked in for the tour": m+ w+ b+ g= 65.
I would put everything in terms of m: w= 3m, g= w- 10= 3m- 10, b= g/4= (3m- 10)/4= 3m/4- 10/4= 3m/4- 5/2.
Then m+ w+ b+ g= m+ 3m+ 3m- 10+ 3m/4- 5/2= 65
(1+ 3+ 3+ 3/4)m= 65+ 10+ 5/2
(4+ 12+ 12+ 3)m/4= 155/2
31m= 310 so m= 10.
Then w= 3m= 30, g= 3m- 10= 20, and b= 3m/4- 5/2= 30/4- 5/2= 15/2- 5/2= 10/2= 5.
There were 10 men, 30 women, 5 boys, and 20 girls.
Check:
"Three times as many women as men are on the tour": yes, 30= 3(10)
"There are 10 less girls than the number of women": yes, 20= 30- 10.
"The number of boys is a quarter of the number of girls": yes, 5= 20/4.
"There are 65 people booked in for the tour": yes, 10+ 30+ 20+ 5= 65.
(Checking is important! The first time I did this problem, I accidently swapped two numbers and only caught the error after checking.)