let y is equal to the least number

then y+5x= 24.5
and y+ 15x= 109.5

then i solved the equation. any shortcut??

Here are two ways of doing it, I don't know whether one's a shortcut or not, but it might help.

First we are given that the difference in price between any larger and next smaller cake is x.
We are also told that largest cake is 24.5 and that all cakes' prices sum is 109.50.

First way:
Largest cake = 24.5
Next smaller cake = 24.5 - x
Next one = 24.5-2x
Next = 24.5-3x
etc.
Smallest is 24.5-5x

You could add all of those and solve for x
24.5*6 - x (1+2+3+4+5) = 109.50 , solving x= 2.5

Another way:
Since the difference of price between the cakes is constant, the cake price is an arithmetic progression.

Now use the formula for sum of a GP: S = n/2 *(2a + (n-1)d) where n is the total number of terms, a is the first term, and d is the constant difference between the terms.
We are given that S = 109.50 ; a=24.50 and d=-x

then 109.50 = 6/2 (2*24.5 - 5x) => 109.50*2/6 = 49 - 5x => 219/2 - 49 = -5x => 25/2 = 5x => x = 2.5

Let me know if this helps

nafiul9090 wrote:A certain bakery sells six di¤erent-sized wedding cakes. Each
cake costs x dollars more than the next one below it in size, and
the price of the largest cake is $24.50. If the sum of the prices of
the six di¤erent cakes is $109.50, what is the value of x?
(A) 1.50
(B) 1.75
(C) 2.00
(D) 2.50
(E) 3.00


is there any short cut method??

The prices are evenly spaced values.
With evenly spaced values, median = average = sum/number.
Thus, the median price of the cakes = (109.5)/6 = 18.25.

We can plug in the answer choices, which represent the distance between consecutive prices.
The median of 18.25 must be halfway between the 3rd greatest price and the 4th greatest price.

Answer choice C: 2
4th greatest price = 24.5 - 2(2) = 20.5, 3rd greatest price = 20.5 - 2 = 18.5.
18.25 is not halfway between 18.5 and 20.5.
The difference between consecutive prices must be greater.
Eliminate A, B and C.

Answer choice D: 2.5
4th greatest price = 24.5 - 2(2.5) = 19.5, 3rd greatest price = 19.5 - 2.5 = 17.
Success! 18.25 is halfway between 17 and 19.5.

The correct answer is D.