A man buys some shirts and some ties. The shirts cost $7 each and the

Question

A man buys some shirts and some ties. The shirts cost $7 each and the ties cost $3 each. If the man spends exactly $81 and buys the maximum number of shirts possible under these conditions, what is the ratio of shirts to ties?

(A) 5:3

(B) 4:3

(C) 5:2

(D) 4:l

(E) 3:2

(A) 5:3

(B) 4:3

(C) 5:2

(D) 4:l

(E) 3:2

niks18 · Accepted Answer

Method 1:  algebraic

let number of shirts be  x  & number of ties be  y  and we need  \frac{x}{y}  i.e.  x:y

so we have  7x+3y=81 ---------(1)

3y=81-7x =>y=\frac{(81-7x)}{3} => y= 27-\frac{7x}{3}

For  y  to be integer  x  has to be a multiple of  3 . so pairs of possible  x  &  y  are

(3,20), (6,13), (9,6).

Now we need maximum shirt so  x=9  &  y=6

Hence ratio =\frac{9}{6}=3:2

Option  E

----------------------------------------------
 Method 2:  use substitution. Work through options.

For  option E:  let shirts be  3x  & tie be  2x

so we have  3x*7+2x*3=81 =>x=3

For other options you will not get the common factor as integer, hence can be rejected.

rahularman · Answer

A basic equation can be formed as below...no of shirts be S, no of ties be T
7S+3T=81
=> 7S=3(27-T)
=> 27-T has t be multiple of 7. To maximize S, minimise T ,

Minimum T cab be 6 so that 27-T ie, 21 is multiple of 7.

=> S=9 & T=6
=> S:T = 3:2

E

JeffTargetTestPrep · Answer

We can create the equation:

7S + 3T = 81

7S = 81 - 3T

7S = 3(27 - T)

S = 3(27 - T)/7

The greatest value of S will be obtained when T is as small as possible. The smallest value of T which produces a number divisible by 7 after substitution is 6, so when T is 6 we have:

S = 3(21)/7 = 9

So, the ratio of S to T is 9 : 6 = 3 : 2.

Answer: E

EgmatQuantExpert · Answer

Solution

Given  
 • A man buys some shirts and some ties.
• Shirts cost $7 each and ties cost $3 each.

To find  
 • The ratio of shirts to ties if the man buys maximum number of shirts possible and spends $81.

Approach and Working out  
 Let the man buys x shirts and y ties then the total spending by the man = 7x + 3y
• So, 7x + 3y = 81
 o Now, if x is maximum then y has to be minimum.

• 7x = 81-3y
 o So, 81- 3y must be a multiple of 7 and y will be minimum.
o For y= 1, 81-3y = 78 -> not a multiple of 7
o For y= 2, 81-6 = 75 -> not a multiple of 7
o For y= 3, 81-9 = 72 -> not a multiple of 7
o For y= 4, 81-12 = 69 -> not a multiple of 7
o For y= 5, 81-15 = 66 -> not a multiple of 7
o For y= 6, 81-18 = 63 -> A multiple of 7 
• x = 9 and y=6
• x: y = 3: 2

Hence, option E is the correct answer.

Correct Answer: Option E

analytica233 · Answer

7S + 3T = 81 --> 3T = 81 - 7S --> T = 27 - (7S)/3.
Now, because T must be an integer, S must be a multiple of 3. Also, we have the maximum of S = 11 (because 7.12 = 84 > 81) --> S = 9 --> T = 27 - 21= 6.
So S:T = 9:6 = 3:2 (E)

rishab0507 · Answer

As total money spend is 81, and person buy max shirts, so we can start calculating with highest shirts possible less than $81 with tie,
Lowest multiple of 7 less than 81 is 77, so shall start with that.
 7*11 + tie = 81, 77+3 < 81, so 11 not possible, as tie value is 3, $1 is left
 7*10 + tie = 81 , 70+ 3*3 < 81, $2 is left, so 10 cannot be no of shirts

7*9 + tie = 81 , tie = 81-63 =18, 
Tie value $3, So total tie would be = 18/3 =6
Hence shirts can be 9 and tie 3
9/3 = 3:2

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