Marnie collects red, blue and purple buttons. At the beginning of the year she has 360 buttons in a box. The ratio of red buttons to blue buttons is 4:3 and there are twice as many red buttons as purple buttons.
a.What is the ratio of blue buttons to purple buttons at the beginning of the year?
b.After Marnie adds another 200 buttons –some red and some blue – at the end of the year, the number of red buttons increased by 75% and the number of blue buttons increased by two-thirds.
i.What is the new ratio of red buttons to blue buttons?
ii.Express the number of purple buttons as a fraction of the new total number of buttons.
iii.Express the number of blue buttons as a percentage of the number of red buttons. (Round to one decimal place).
so, we have red:blue:purple = 4:3:2
4+3+2=9, so there are 360/9 buttone for each proportion:
160 red
120 blue
80 purple
b:p = 3:2
after new buttons, we have
red: 160 (1.75) = 280
blue: 120(5/3) = 200
purple: 80
r:b = 280:200 = 7:5
p/all = 80/560 = 1/7
b/r = 200/280 = 5/7 = 71.4%
I'm confused on where you got the 1.75 and the 5/3????
To solve this problem, we'll break it down into steps:
Step 1: Find the number of red buttons, blue buttons, and purple buttons at the beginning.
Step 2: Determine the ratio of blue buttons to purple buttons.
Step 3: Calculate the new number of red and blue buttons after Marnie adds 200 more buttons.
Step 4: Find the new ratio of red buttons to blue buttons.
Step 5: Express the number of purple buttons as a fraction of the new total number of buttons.
Step 6: Express the number of blue buttons as a percentage of the number of red buttons.
Now let's go through each step:
Step 1: Find the number of red buttons, blue buttons, and purple buttons at the beginning.
Given that the ratio of red buttons to blue buttons is 4:3 and there are twice as many red buttons as purple buttons, we can set up the following equations:
4x + 3x + 2x = 360,
where x is the number of purple buttons.
Simplifying the equation, we get:
9x = 360,
x = 360 / 9 = 40.
So, there are 4x = 4 * 40 = 160 red buttons, 3x = 3 * 40 = 120 blue buttons, and 2x = 2 * 40 = 80 purple buttons at the beginning.
Step 2: Determine the ratio of blue buttons to purple buttons.
The ratio of blue buttons to purple buttons at the beginning is given by 120:80, which can be simplified to 3:2.
Step 3: Calculate the new number of red and blue buttons after Marnie adds 200 more buttons.
The number of red buttons increases by 75%, which means the number of red buttons increases by (75/100) * 160 = 120.
The number of blue buttons increases by two-thirds, which means the number of blue buttons increases by (2/3) * 120 = 80.
So, the new number of red buttons is 160 + 120 = 280, and the new number of blue buttons is 120 + 80 = 200.
Step 4: Find the new ratio of red buttons to blue buttons.
The new ratio of red buttons to blue buttons is 280:200, which can be simplified to 7:5.
Step 5: Express the number of purple buttons as a fraction of the new total number of buttons.
The new total number of buttons is 360 + 200 = 560.
The number of purple buttons is still 80.
So, the number of purple buttons as a fraction of the new total number of buttons is 80/560 = 1/7.
Step 6: Express the number of blue buttons as a percentage of the number of red buttons.
The number of blue buttons is 200 and the number of red buttons is 280.
To express the number of blue buttons as a percentage of the number of red buttons, we divide the number of blue buttons by the number of red buttons and multiply by 100:
(200/280) * 100 = 71.43%.
So, the number of blue buttons is approximately 71.43% of the number of red buttons.