Question

There were 820 orange buttons in a container. The number of orange buttons was 160 fewer than the number of yellow buttons and 210 more than the number of green buttons. 1/3 of the total number of buttons in the container were blue buttons. How many buttons were there altogether?

Answer 1

Answer: There are 3615 buttons

Step-by-step explanation:

From this situation we can write a system of equations if we tag the orange buttons with $o$, the yellow buttons with $y$, the green buttons with $g$ and the blue buttons with $b$:

There were 820 orange buttons:

$o=820$ (1) Number of orange buttons

The number of orange buttons was 160 fewer than the number of yellow buttons:

$o=y-160$ (2)

The number of orange buttons was 210 more than the number of green buttons:

$o=g+210$ (3)

1/3 of the total number of buttons in the container were blue buttons:

If the total is the sum of the buttons of each color, we have:

$\frac{1}{3}(o+y+g+b)=b$ (4)

At this point we have our system with 4 equations and 4 unknowns.

Let's begin by substituting (1) in (2):

$820=y-160$ (5)

Isolating $y$:

$y=980$ (6) Number of yellow buttons

Subsituting (1) in (3):

$820=g+210$ (7)

Isolating $g$:

$g=610$ (8) Number of green buttons

Substituting (1), (6) and (8) in (4):

$\frac{1}{3}(820+980+610+b)=b$ (9)

Isolating $b$:

$b=1205$ (10) Number of blue buttons

Now we can find the total number of buttons:

$o+y+g+b=820+980+610+1205=3615$ (11) This is the total number of buttons