A group of people went to a coffee shop.

Each person chose a drink and a treat.


number of people who chose coffee: number of people who chose tea = 1:4


Of those who chose coffee,

number of people who chose a muffin number of people who chose a tart = 2:7


Of those who chose tea,

number of people who chose a muffin : number of people who chose a tart = 5:1


What fraction of the people chose a tart?

Answer:

Fraction of the people chose a tart = [tex]\frac{13}{45}[/tex]

Step-by-step explanation:

Given - A group of people went to a coffee shop. Each person chose a

            drink  and a treat. number of people who chose coffee: number of  

            people who chose tea = 1:4. Of those who chose coffee,  number of  

            people who chose a muffin number of people who chose a tart =    

            2:7. Of  those who chose tea,  number of people who chose a muffin  

            : number of people who chose a tart = 5:1.

To find - What fraction of the people chose a tart?

Proof -

Let the Total number of People went to a coffee shop = 5x

Now,

As given, number of people who chose coffee: number of people who chose tea = 1:4

⇒Total number of people who chooses coffee = [tex]\frac{1}{5}.5x[/tex] = x

  Total number of people who chooses tea  = [tex]\frac{4}{5}.5x[/tex] = 4x

Now,

As given,

Of those who choose coffee,  number of people who chose a muffin number of people who chose a tart = 2:7

⇒Total number of people who choose muffin after coffee = [tex]\frac{2}{9}x[/tex]

   Total number of people who choose tart after coffee = [tex]\frac{7}{9}x[/tex]

Also,

Given that

Of those who chose tea,  number of people who chose a muffin : number of people who chose a tart = 5:1

⇒Total number of people who choose muffin after tea = [tex]\frac{5}{6}.4x = \frac{20}{6}x[/tex]

   Total number of people who choose tart after tea = [tex]\frac{1}{6}.4x = \frac{4}{6}x[/tex]

∴ we get

Total number of people who chooses tart = Total number of people who choose tart after coffee + Total number of people who choose tart after tea

⇒Total number of people who chooses tart = [tex]\frac{7}{9}x[/tex] + [tex]\frac{4}{6}x[/tex]

                                                                         = [tex]\frac{7}{9}x + \frac{2}{3}x[/tex]

                                                                         = [tex]\frac{7x + 6x}{9}[/tex]

                                                                         = [tex]\frac{13}{9}x[/tex]

⇒Total number of people who chooses tart = [tex]\frac{13}{9}x[/tex]

Now,

Fraction of the people chose a tart = [tex]\frac{People choose tart}{Total people}[/tex]

                                                           = [tex]\frac{\frac{13}{9}x }{5x}[/tex]

                                                           = [tex]\frac{13}{45}[/tex]

⇒Fraction of the people chose a tart = [tex]\frac{13}{45}[/tex]

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