Two ratios, one shared quantity

In a canteen fridge, the number of bottles of water to the number of cartons of milk is 3 to 4. The number of cartons of milk to the number of cans of juice is 6 to 5. There are 45 bottles of water in the fridge. How many drinks are there in the fridge altogether? Try it first. Pause if you like. Start with the three drinks. Water, milk and juice, each with a column of its own. The milk is in the middle, because both ratios count it. Both ratios are about the milk, so write them one under the other. Water to milk is 3 to 4. Milk to juice is 6 to 5. The milk is a 4 in the first line and a 6 in the second. To put all three on one line, the milk has to be the same number in both. 4 and 6 both go into 12. 12 is the lowest common multiple of 4 and 6. Now multiply each ratio up to 12. Multiply the first line by 3, and 3 to 4 becomes 9 to 12. Multiply the second line by 2, and 6 to 5 becomes 12 to 10. Now the milk column says 12 in both lines. Now all three fit on one line. Water to milk to juice is 9 to 12 to 10. The question gives us the water. 9 units is 45. So 1 unit is 45 divided by 9. That's 5. Add the three terms. 9 and 12 and 10 is 31 units. 31 times 5 is one hundred and fifty five. That's how many drinks are in the fridge. One quantity, counted twice. Make the milk the same number of units in both ratios. Then every unit means the same amount, and the three terms fit on one line.