A factory produced a total of 850 pink and blue hairclips. 1/3 of the pink hairclips and 100 of the blue hairclips were sold. There was an equal number of pink and blue hairclips left in the end. Express the ratio of the number of pink hairclips to blue hairclips at first.
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b + p = 850 so b = 850 - p
(2/3) p = b-100
(2/3) p = 850 - p - 100
(2/3)p = 750 - p
(5/3) p = 750
p = 3 * 150 = 450
so b = 400
p/b = 450/400 = 9/8
9:8 is the ratio of pink hairclips to blue hairclips at first
Let's work through the problem step-by-step.
Step 1: "A factory produced a total of 850 pink and blue hairclips."
Let's assume the number of pink hairclips produced is P, and the number of blue hairclips produced is B. According to the given information, we know that P + B = 850.
Step 2: "1/3 of the pink hairclips and 100 of the blue hairclips were sold."
If 1/3 of the pink hairclips were sold, the number of pink hairclips left is 2/3P. Similarly, the number of blue hairclips left is B - 100.
Step 3: "There was an equal number of pink and blue hairclips left in the end."
Since the number of pink hairclips left is equal to the number of blue hairclips left, we can set up an equation: 2/3P = B - 100.
Step 4: Express the ratio of the number of pink hairclips to blue hairclips at first.
To express the ratio of pink hairclips to blue hairclips at first, we need to find the values of P and B. To do this, we can substitute P + B = 850 into the equation in step 3:
2/3P = B - 100
Now, substitute P = 850 - B into the equation:
2/3(850 - B) = B - 100
Multiply both sides of the equation by 3 to eliminate the fraction:
2(850 - B) = 3(B - 100)
2(850 - B) = 3B - 300
1700 - 2B = 3B - 300
Add 2B to both sides and add 300 to both sides:
2000 = 5B
Divide both sides by 5:
B = 400
Now, substitute B = 400 into P + B = 850:
P + 400 = 850
P = 850 - 400
P = 450
The initial ratio of pink hairclips to blue hairclips is P:B = 450:400, which simplifies to 9:8.
Therefore, the ratio of the number of pink hairclips to blue hairclips at first is 9:8.
To find the ratio of the number of pink hairclips to blue hairclips at first, we need to determine the initial number of pink and blue hairclips before any were sold.
Let's represent the number of pink hairclips as "P" and the number of blue hairclips as "B".
According to the problem, we know that a total of 850 hairclips were produced, so we can write an equation based on this information:
P + B = 850
We are also given two additional pieces of information: 1/3 of the pink hairclips were sold, and 100 of the blue hairclips were sold.
To find the total number of pink hairclips sold, we need to calculate 1/3 of the initial number of pink hairclips:
Pink hairclips sold = (1/3) * P
To find the total number of blue hairclips sold, we subtract 100 from the total number of blue hairclips:
Blue hairclips sold = B - 100
The number of pink and blue hairclips left in the end is stated to be equal, so:
(P - Pink hairclips sold) = (B - Blue hairclips sold)
Substituting the expressions for Pink hairclips sold and Blue hairclips sold, we get:
P - (1/3) * P = B - (B - 100)
Simplifying further:
(2/3) * P = 100
To isolate P, we multiply both sides of the equation by 3/2:
P = (3/2) * 100
Simplifying gives:
P = 150
Substituting this value back into the equation for P + B = 850:
150 + B = 850
B = 850 - 150
B = 700
So, there were 150 pink hairclips and 700 blue hairclips initially.
The ratio of the number of pink hairclips to blue hairclips at first is 150:700, which simplifies to 3:14.
b + p = 850 so b = 850 - p
(2/3) p = b-100
(2/3) p = 850 - p - 100
(2/3)p = 750 - p
(5/3) p = 750
p = 3 * 150 = 450
so b = 400
p/b = 45/40 = 1.125