Combined Ratios

“There are red, blue and green balls in a jar. The ratio of red to blue balls is 9 : 7 and the ratio of blue  to green balls is 3 : 11. What is the smallest possible number of balls in the jar?”

This question was given not only in the Open category but in all other age groups except the beginners. The results show than only 25 to 30% from the 5 groups got the correct answer to this question which can be solved mentally.

Two ratios are given, and the problem requires the least possible number of balls in the jar.  

The number of blue balls is common in the two ratios, so we will put it as the middle value in the combined ratios.

We have the ratio of red balls to blue is 9:7 and the blue to green balls  is  3: 11. Therefore, least number of blue balls is the least common multiple of 7 and 3 which is 21.

If there are 21 blue balls, then we should also multiply the number of green balls by 7 to get 77, making the total of blue and green balls to 98.

Then we also have to multiply the number of red balls, 9, by 3 to get 27 for a final sum of  125.