2/7 of the number of roses in Angel Florist was equal to 3/4 of the number of roses in Beauty Florist at first. After Angel Florist sold 84 roses and Beauty Florist bought another 58 roses, the ratio of the number of roses in Angel Florist to the number of roses in Beauty Florist became 3:2. What was the total number of roses in Angel Florist and Beauty Florist at first?
original:
number in Angel's place --- a
number in Beauty's place --- b
(2/7)a = (3/4)b
8a = 21b or a = 21b/8
after transactions:
number for Angle = a - 84
number for Beauty = b + 58
(a-84)/(b+58) = 3/2
2a - 168 = 3b + 174
but from earlier: a = 21b/8
2(21b/8) - 168 = 3b + 174
21b/4 = 3b + 342
times 4
21b = 12b + 1368
9b = 1368
b = 152 , then a = 21(152)/8 = 399
To solve this problem, let's start by assigning variables to the unknown quantities. Let's say the number of roses in Angel Florist is "A" and the number of roses in Beauty Florist is "B."
According to the information given, we have two statements:
1) "2/7 of the number of roses in Angel Florist was equal to 3/4 of the number of roses in Beauty Florist at first."
This can be written as the equation: (2/7)A = (3/4)B.
2) "After Angel Florist sold 84 roses and Beauty Florist bought another 58 roses, the ratio of the number of roses in Angel Florist to the number of roses in Beauty Florist became 3:2."
This can be written as the equation: (A - 84) / (B + 58) = 3/2.
We can solve this system of equations to find the values of A and B.
Step 1: Solve the first equation for A
Multiply both sides of the equation by (7/2) to get:
A = (7/2)(3/4)B
A = (21/8)B
Step 2: Substitute the value of A from the first equation into the second equation
[(21/8)B - 84] / (B + 58) = 3/2
Step 3: Solve for B
Multiply both sides of the equation by 2(B + 58) to get rid of the fractions:
2[(21/8)B - 84] = 3(B + 58)
Simplify the equation:
(21/4)B - 168 = 3B + 174
Combine like terms:
(21/4)B - 3B = 174 + 168
(21B - 12B) / 4 = 342
9B = 1368
B = 1368/9
B = 152
Step 4: Substitute the value of B back into the first equation to find A
A = (21/8)B
A = (21/8)(152)
A = 399
Therefore, the total number of roses in Angel Florist and Beauty Florist at first is A + B = 399 + 152 = 551.
2/7 a = 3/4 b
(a-84)/(b+58) = 3/2
now solve as usual, then figure a+b