A juice mix uses 2 cups of concentrate and 6 cups of water. Step 1 (the additive description): water is "4 more cups than concentrate." True - and useless for scaling. Step 2 (the multiplicative description): 6 = 3 × 2, so water is 3 × the concentrate. Turn it around: 2 = (1/3) × 6, so concentrate is 1/3 × the water. Either quantity can be described as a multiple of the other - that is the curriculum's definition of a multiplicative comparison. Step 3 (why it wins): scale the batch up to 5 cups of concentrate. The multiple predicts water = 3 × 5 = 15 cups. The additive description predicts 5 + 4 = 9 cups - a much weaker drink. Doubling "4 more" fails the same way (4 and 10): the DIFFERENCE survives adding, but only the MULTIPLE survives scaling. Say it plainly: a ratio is a multiplication fact wearing a colon. When someone hands you two quantities, your first question is "how many times?" - not "how much more?"

A ribbon display uses 5 m of red ribbon and 15 m of blue ribbon. Describe each quantity as a multiple of the other. -> Blue is 3 × the red (15 = 3 × 5); red is 1/3 × the blue (5 = (1/3) × 15).