Question

The ratio of the number of yellow buttons that Miko had to the number of blue buttons that Miko had was 2: 3. After she had bought 27 more blue buttons, The ratio of the number of yellow buttons to the number of blue buttons became 4:9. How many buttons did Miko have at first?​

Answer 1

Let's assume the original number of blue buttons Miko had was "b".

From the first ratio, we know that for every 3 blue buttons, Miko had 2 yellow buttons, so the original number of yellow buttons was (2/3)b.

When Miko bought 27 more blue buttons, the number of blue buttons became b + 27. The ratio of yellow buttons to blue buttons after Miko bought the extra buttons became 4:9, so the number of yellow buttons became (4/9)(b + 27).

We can set these two expressions for the number of yellow buttons equal to each other and solve for b:

(2/3)b = (4/9)(b + 27)

Expanding the right-hand side:

(2/3)b = (4/9)b + (4/9)(27)

Combining like terms:

(2/3)b - (4/9)b = (4/9)(27)

Solving for b:

(5/9)b = (4/9)(27)

Dividing both sides by (5/9):

b = 27

So Miko originally had b = 27 blue buttons. To find the original number of yellow buttons, we can use the equation:

y = (2/3)b = (2/3)(27) = 18

So Miko originally had y = 18 yellow buttons

Answer 2

90 buttons

Explanation:

the ratio of yellow : blue = 2 : 3 = 2x : 3x ( x is a multiplier )

after buying 27 more blue buttons

2x : 3x + 27 = 4 : 9

expressing the ratios in fractional form

`(2x)/(3x+27)` =
`(4)/(9)` ( cross- multiply )

18x = 4(3x + 27)

18x = 12x + 108 ( subtract 12x from both sides )

6x = 108 ( divide both sides by 6 )

x = 18

number of buttons at first = 2x + 3x = 5x = 5 × 18 = 90