I’m prepping for a test and trying to lock down proportional reasoning, but I keep second-guessing my setup. Suppose a lemonade recipe is 2 parts concentrate to 7 parts water (so total parts = 9). If I want a total of 4.5 liters, I set it up like: let each “part” be x liters, so concentrate = 2x, water = 7x, total = 9x. Then 9x = 4.5, so x = 4.5/9, and concentrate should be 2x. That feels neat because the 2:7 pattern scales cleanly.
But then I get tripped up when the problem changes slightly. If I’ve already poured 1.2 liters of water and I still want to keep the 2:7 ratio, I thought: water = 7x = 1.2, so x = 1.2/7, and concentrate should be 2x. Alternatively, I tried using a constant of proportionality: c/w = 2/7, so c = (2/7)w. Those seem equivalent, but I’m worried I’m flipping something without noticing.
Are both approaches (parts method vs. using c/w = 2/7) basically the same and acceptable on a test? And in the “already poured water” version, is setting 7x = 1.2 the right starting point, or should I be anchoring to total volume instead? I love how the ratio acts like a scaled vector, but I don’t want to overcomplicate it right before the test. Any tips on a reliable way to set this up so I don’t mix up which quantity goes on top?
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3 Responses
Yep-both methods are the same scaling idea; with 1.2 L of water you can set 7x = 1.2 (so x = 1.2/7), then c = 2x = (2/7)·1.2, and you only use 9x if the total is what’s given.
Analogy: think 2 red + 7 blue Lego bricks all stretched by the same amount; to avoid flipping, I always label it “c/w = 2/7” (concentrate on top, water below).
I’m right there with you-ratios are like a recipe’s blueprint or a Lego build: “2 red bricks for every 7 blue,” and everything just scales by one common factor. Your first setup is perfect: with total 4.5 L, take k = 4.5/9 = 0.5 L per part, so concentrate = 2k = 1.0 L and water = 7k = 3.5 L. The two approaches you described (parts method vs. c/w = 2/7) are exactly the same under the hood: both say “concentrate = 2k, water = 7k” for some scale k. In the “already poured water” version, setting 7k = 1.2 is the right move if the only constraint is “match the ratio now,” giving k = 1.2/7 and concentrate = 2k = (2/7)·1.2 ≈ 0.343 L, total ≈ 1.543 L. If you also have a target total (say still 4.5 L), then anchor to the total: aim for water = (7/9)·4.5 = 3.5 L and concentrate = (2/9)·4.5 = 1.0 L; since you’ve already got 1.2 L of water, you’d add 2.3 L more water and 1.0 L concentrate. A foolproof habit: write the ratio with labels and stick to the same order-c:w = 2:7 implies c/w = 2/7, so c = (2/7)w and w = (7/2)c; or use the “one k to rule them all” rule: component = (its part)·k, and determine k from whatever quantity is given (total or a component). Which anchor feels more natural to you in test mode-locking onto k from the total, or using c = (2/7)w from a known component?
Hey! I really like how you’re thinking about this. The “parts” idea and the “constant of proportionality” idea are two faces of the same coin, and you’re using them in a very sane way.
Big picture:
– Parts method: set concentrate = 2x and water = 7x, so total = 9x.
– Proportion method: c/w = 2/7, so c = (2/7)w (and equivalently w = (7/2)c).
Those two are equivalent because if c = 2x and w = 7x, then c/w = (2x)/(7x) = 2/7. And going the other way, if c/w = 2/7, you can let x = w/7 so that w = 7x and c = 2x. So either form is totally fine on a test.
Working your two scenarios:
1) Target total = 4.5 L
– 9x = 4.5, so x = 4.5/9 = 0.5 L per part.
– Concentrate = 2x = 1.0 L.
– Water = 7x = 3.5 L.
You can also think in “fraction of total” terms: concentrate is 2/9 of the total and water is 7/9 of the total. A quick mental check: 2/9 of 4.5 is 0.9… wait, I always misread that; 2/9 of 4.5 is actually 1.0. That matches the parts method nicely.
2) Already poured 1.2 L of water, keep 2:7
– If the only fixed thing is that water you already poured (and you don’t care what the final total becomes), then anchoring with 7x = 1.2 is exactly right:
– x = 1.2/7 ≈ 0.1714286
– Concentrate = 2x ≈ 0.342857 L
– Total mix will be about 1.542857 L.
– If instead you still want a specific total (say you still want 4.5 L), then anchor to the total:
– 9x = 4.5 ⇒ x = 0.5
– Target water = 7x = 3.5 L; target concentrate = 2x = 1.0 L.
– You already have 1.2 L of water, so add 3.5 − 1.2 = 2.3 L more water, and add 1.0 L concentrate.
Both ways are acceptable on a test. The “right” starting point depends on what’s actually fixed: if the water amount is fixed, use 7x = (that water); if the total is fixed, use 9x = (that total). If both are fixed, check consistency (for example, if total is 4.5 L, the water must be 7/9 of that, i.e., 3.5 L-so starting with 1.2 L is fine because you can still add up to 3.5 L).
A couple of habits to avoid flipping the ratio:
– Keep the order consistent. If you say “concentrate : water = 2 : 7,” then when you turn it into a fraction, make it c/w = 2/7 (not 7/2). I still sometimes write 2c = 7w by accident when cross-multiplying; the correct cross-multiplication from c/w = 2/7 is 7c = 2w.
– Use the “fraction of total” view when the total is given: concentrate is 2/9 of total, water is 7/9 of total. I like this because it auto-checks that the two fractions add to 1.
– Sanity check with decimals: c/w = 2/7 ≈ 0.2857. That means concentrate is a bit less than a third of the water. If your numbers don’t feel like that, something’s flipped.
If you want a clean refresher on ratios and setting up proportions, this walkthrough is solid:
https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-ratios-prop-topic/cc-6th-ratios/a/intro-to-ratios
You’ve got the right idea-just decide what’s fixed (total or one component), write the matching equation (9x = total, or 7x = water), and the rest falls into place.