Answer:
The skirts left in the store are <u>260</u> and blouses left are<u> 270</u>.
Step-by-step explanation:
Given:
An apparel shop sells skirts for $45 and blouses for $35.
The worth of entire stock is $51,750.
Now, to find how many skirts and blouses are left in the store.
Let the total price of skirts be ![x.](https://tex.z-dn.net/?f=x.)
And let the total price of blouses be ![y.](https://tex.z-dn.net/?f=y.)
So, price of the entire stock is:
![x+y=51,750\\\\y=51750-x \ \ \ \ .....(1)](https://tex.z-dn.net/?f=x%2By%3D51%2C750%5C%5C%5C%5Cy%3D51750-x%20%5C%20%5C%20%5C%20%5C%20.....%281%29)
As, half the skirts and two-thirds of the blouses are sold, for a total of $30,600.
Now, the price of the half skirts and two-thirds of the blouses:
![\frac{1}{2} x+\frac{2}{3} y=30,600\\\\\frac{x}{2} +\frac{2y}{3} =30,600](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20x%2B%5Cfrac%7B2%7D%7B3%7D%20y%3D30%2C600%5C%5C%5C%5C%5Cfrac%7Bx%7D%7B2%7D%20%2B%5Cfrac%7B2y%7D%7B3%7D%20%3D30%2C600)
Substituting the value of
from equation (1) we get:
![\frac{x}{2}+\frac{2\times(51750-x) }{3}=30,600\\\\\frac{x}{2}+\frac{(103500-2x) }{3}=30,600\\\\\frac{ 3x+207000-4x}{6}=30,600](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B2%7D%2B%5Cfrac%7B2%5Ctimes%2851750-x%29%20%7D%7B3%7D%3D30%2C600%5C%5C%5C%5C%5Cfrac%7Bx%7D%7B2%7D%2B%5Cfrac%7B%28103500-2x%29%20%7D%7B3%7D%3D30%2C600%5C%5C%5C%5C%5Cfrac%7B%203x%2B207000-4x%7D%7B6%7D%3D30%2C600)
<em>Multiplying both sides by 6 we get:</em>
<em />
<em />
<em />
<em />
<em>Subtracting both sides by 207000 we get:</em>
<em />
<em />
<u><em>The total price of skirts = $23,400.</em></u>
By substituting the value of
in equation (1) we get:
![y=51750-x\\\\y=51750-23400\\\\y=28350.](https://tex.z-dn.net/?f=y%3D51750-x%5C%5C%5C%5Cy%3D51750-23400%5C%5C%5C%5Cy%3D28350.)
<u><em>The total price of blouses = $28,350.</em></u>
The sale price of a skirt = $45.
The sale price of a blouse = $35.
Now, to get the total number of skirts and blouses of the stock:
The total number of skirts in the stock = The total price of skirts ÷ the sale price of a skirt.
![=23,400\div 45\\\\=520.](https://tex.z-dn.net/?f=%3D23%2C400%5Cdiv%2045%5C%5C%5C%5C%3D520.)
<em><u>The total number of skirts in the stock = 520.</u></em>
The total number of blouses in the stock = The total price of blouses ÷ the sale price of a blouse.
![=28350\div 35\\\\= 810.](https://tex.z-dn.net/?f=%3D28350%5Cdiv%2035%5C%5C%5C%5C%3D%20810.)
<u><em>The total number of blouses in the stock = 810.</em></u>
<em>In the shop only half of the skirts and two-thirds of the blouses are sold.</em>
Now, to get the skirts left in the store:
![520-\frac{1}{2}\ of\ 520 \\\\=520-0.5\times 520\\\\=520-260\\\\=260.](https://tex.z-dn.net/?f=520-%5Cfrac%7B1%7D%7B2%7D%5C%20of%5C%20520%20%5C%5C%5C%5C%3D520-0.5%5Ctimes%20520%5C%5C%5C%5C%3D520-260%5C%5C%5C%5C%3D260.)
<u><em>Thus, 260 skirts are left in the store.</em></u>
Now, to get the blouses left in the store:
![810-\frac{2}{3} \ of\ 810\\\\=810-\frac{2}{3} \times 810\\\\=810-\frac{1620}{3} \\\\=810-540\\\\=270.](https://tex.z-dn.net/?f=810-%5Cfrac%7B2%7D%7B3%7D%20%5C%20of%5C%20810%5C%5C%5C%5C%3D810-%5Cfrac%7B2%7D%7B3%7D%20%5Ctimes%20810%5C%5C%5C%5C%3D810-%5Cfrac%7B1620%7D%7B3%7D%20%5C%5C%5C%5C%3D810-540%5C%5C%5C%5C%3D270.)
<em><u>Hence, 270 blouses are left in the store.</u></em>
Thus, the skirts left in the store are 260 and blouses left are 270.