By increasing the cost of entry to a fair in the ratio 10:13, the number of visitors to the fair has decreased in the ratio 6:5. In what ratio has the total collection increased or decreased ? Solve using two different method.

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Solution

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$Method 1 : \frac{O r i g i n a l T i c k e t}{I n c r e a s e d T i c k e t} = \frac{10}{13} \frac{O r i g i n a l V i s i t o r s}{F i n a l V i s i t o r s} = \frac{6}{5} C o l l e c t i o n R a t i o s = \frac{O r i g i n a l T i c k e t \times O r i g i n a l V i s i t o r s}{I n c r e a s e d T i c k e t \times F i n a l V i s i t o r s} = \frac{O r i g i n a l T i c k e t}{I n c r e a s e d T i c k e t} \times \frac{O r i g i n a l V i s i t o r s}{F i n a l V i s i t o r s} = \frac{10}{13} \times \frac{6}{5} = \frac{12}{13} C o l l e c t i o n R a t i o = \frac{o r i g i n a l c o l l e c t i o n}{F i n a l c o l l e c t i o n} = \frac{12}{13} T h e r e f o r e t h e r e w i l l b e a i n c r e a s e i n t h e c o l l e c t i o n . M e t h o d 2 : L e t t h e o r i g i n a l p r i c e b e 10 x a n d i n c r e a s e d p r i c e b e 13 x l e t t h e i n i t i a l n u m b e r o f V i s i t o r s b e 6 y a n d f i n a l n u m b e r o f v i s i t o r s b e 5 y, O r i g i n a l c o l l e c t i o n = o r i g i n a l p r i c e \times i n i t i a l n u m b e r o f V i s i t o r s = 10 x \times 6 y = 60 x y F i n a l c o l l e c t i o n = i n c r e a s e d p r i c e \times f i n a l n u m b e r o f V i s i t o r s = 13 x \times 5 y = 65 x y C o l l e c t i o n R a t i o = \frac{O r i g i n a l c o l l e c t i o n}{F i n a l c o l l e c t i o n} = \frac{60 x y}{65 x y} = \frac{12}{13} R e g a r d s$

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