Alice and Bernice shared some stickers in the ratio 3:1. Alice had 2/3 as many stickers as Candice. Candice had 21 more stickers than Bernice.
(a) What is the ratio of the number of stickers Alice had to the number of stickers Bernice had to the number of stickers Candice had?
(b) How many stickers did they have altogether?
a = 3b
a = 2/3 c
c = b+21
a:b:c = 18:6:27
Let's solve this step by step:
(a) To find the ratio of the number of stickers Alice had to the number of stickers Bernice had to the number of stickers Candice had, we need to find the number of stickers each person had.
Let's assume that Alice had 3x stickers, Bernice had x stickers, and Candice had y stickers.
Given information:
Alice and Bernice shared stickers in the ratio 3:1, so we can write: Alice's stickers / Bernice's stickers = 3/1. This can be represented as: (3x) / (x) = 3/1.
Alice had 2/3 as many stickers as Candice, so we can write: Alice's stickers / Candice's stickers = 2/3. This can be represented as: (3x) / y = 2/3.
Candice had 21 more stickers than Bernice, so we can write: Candice's stickers - Bernice's stickers = 21. This can be represented as: y - x = 21.
Now we have a system of three equations:
(3x) / (x) = 3/1,
(3x) / y = 2/3,
y - x = 21.
To solve this system, we can use the method of substitution or elimination. Let's solve it using the method of substitution:
From equation 1, we get: x = 3x/3 → x = x. The value of x is not determined from this equation.
From equation 3, we get: y = 21 + x.
Substituting these values in equation 2, we get:
(3x) / (21 + x) = 2/3.
Cross-multiplying, we get:
3(3x) = 2(21 + x).
Simplifying, we get:
9x = 42 + 2x.
Bringing like terms together, we get:
9x - 2x = 42.
Simplifying further, we get:
7x = 42.
Dividing both sides by 7, we get:
x = 6.
Since x represents the number of stickers Bernice had, Bernice had 6 stickers.
From equation 3, we get:
y = 21 + x = 21 + 6 = 27.
Therefore, Bernice had 6 stickers, Alice had 3x = 3(6) = 18 stickers, and Candice had y = 27 stickers.
The ratio of the number of stickers Alice had to the number of stickers Bernice had to the number of stickers Candice had is:
18:6:27, which simplifies to 3:1:9.
(b) To find the total number of stickers they had altogether, we simply add up the number of stickers each person had:
Total number of stickers = Alice's stickers + Bernice's stickers + Candice's stickers
Total number of stickers = 18 + 6 + 27
Total number of stickers = 51.
Therefore, they had 51 stickers altogether.
To answer this question, let's break it down step by step.
(a) First, let's find an equation that represents the relationship between Alice, Bernice, and Candice.
We are given that Alice had 2/3 as many stickers as Candice. Let's represent the number of stickers Alice had as A, and the number of stickers Candice had as C. So we have the equation:
A = (2/3)C
Next, we are given that Candice had 21 more stickers than Bernice. Let's represent the number of stickers Bernice had as B. So we have the equation:
C = B + 21
Now let's look at the stickers shared between Alice and Bernice in a ratio of 3:1. Since the ratio of Alice to Bernice is 3:1, we can represent the number of stickers Alice had as 3x and the number of stickers Bernice had as x, where x is a common factor. So we have the equation:
A = 3x
B = x
Now we can substitute the values of A and B into the equation we found earlier:
C = B + 21
C = x + 21
Since we already have the equation A = (2/3)C, we can substitute the value of C as x + 21:
A = (2/3)(x + 21)
Now we have equations for A, B, and C, and we can find the ratio of the number of stickers Alice had to the number of stickers Bernice had to the number of stickers Candice had:
A : B : C = (3x) : x : (x + 21)
(b) To find the total number of stickers they had altogether, we need to add the number of stickers each person had:
Total = A + B + C
We can substitute the values we found earlier for A, B, and C into this equation:
Total = (3x) + x + (x + 21)
Simplifying this equation will give us the answer to part (b).