25 is the answer.
G = green sweets
Ratio of blue to green = 3:5
Ratios are essentially fractions, and can be converted into fraction form
Therefore, number of blue sweets is 3/5 of green sweets, which can also be represented as 3/5g
Number of green sweets to red sweets is 6:1
We can also flip this ratio around and the ratio would still have the same value.
Number of red sweets to green sweets = 1:6
In fraction form, the number of red sweets is 1/6 of green sweets, which is also 1/6g.
Since there are less than 270 sweets in the box, we can use this expression to represent the sweets in the box:
G + 3/5g + 1/6g < 270
Find a common denominator for the fractions.
5 * 6 = 30, thus a common denominator would be 30.
Since G is a variable with no coefficient, 1 would be implied because no coefficient is present.
So when multiplying G to get a common denominator, the numerator would be 30. 30/30 = 1
G * 30 = 1 * 30
G = 30/30g
Multiply the rest of the numerators and you’ll get:
30/30g + 18/30g + 5/30g < 270
Add.
1 + 18/30g + 5/30g = 1 23/30g
Convert this into an improper fraction.
53/30g < 270
Now divide both sides by 53/30.
(53/30g) ÷ (53/30) < 270 ÷ (53/30)
= g < 270 ÷ (53/30)
Dividing fractions is complex, so you can simply switch the numerator and denominator and multiply:
G < 270 x (30/53)
= g < 8,100/53
Since the number of red sweets to green sweets is 1/6g the number of green sweets, the number of red sweets is 1/6 of 8,100/53.
R = red sweets
R < 8,100/53 x 1/6
8,100 x 1 = 8,100
53 x 6 = 318
R < 8,100/318
318 goes into 8,100 25 times, regardless of any remainder.
Therefore, the greatest possible number of red sweets in the box is 25 red sweets.
