6870
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1
2011-02-01T00:00:01
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<div style="float: right;"><mdo:image alt="" height="292" src="jugs1.png" width="191"></mdo:image></div>
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I mixed up some lemonade in two glasses.<br></br>
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The first glass had $200$ml of lemon juice and $300$ml of water.<br></br>
The second glass had $100$ml of lemon juice and $200$ml of water.<br></br>
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<span style="font-weight: bold;">Which mixture has the stronger tasting lemonade?</span><br></br>
<span style="font-weight: bold;">How do you know?</span><br></br>
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Use the interactivity below to compare different mixtures of lemonade and develop a strategy for deciding which is stronger each time.<br></br>
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<a href="/content/id/6870/lemon2.swf">Full Screen Version</a><br></br>
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<param name="movie" value="/content/id/6870/lemon2.swf"></param>
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<div style="font-weight: bold;"><span style="font-weight: 400;">Once you are confident that you can always work out which mixture is stronger,</span> here are some questions to consider:</div>
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<div>How might you use fractions to help you to work out which mixture is stronger?</div>
<div>How might you use ratios?</div>
<div>How about a graphical approach?</div>
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<div>Do you always use the same strategy?</div>
<div>Describe some occasions when one strategy might be more efficient than another.</div>
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<div>In the original example, the first glass had 200ml of lemon juice and 300ml of water, and the second glass had 100ml of lemon juice and 200ml of water. If I mix the two glasses of lemonade together, the mixture is weaker than the first glass was, but stronger than the second glass.</div>
<div>Try the same with some other mixtures. Is the strength of the combined mixture always between the strengths of the originals? Can you justify your findings?</div>
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<p class="editorial">Lots of great solutions were submitted to this
problem, using a wide variety of approaches. The problem prompted
you to use fractions, ratios, percentages, and graphs. In addition,
you could investigate and consider which methods worked most
effectively in different situations.</p>
<p><span class="editorial">The first part of the task is to
determine the mixture with the stronger tasting lemonade
-</span></p>
<div style="margin-left: 40px;"><span class="editorial">the first
glass has $200$ml of lemon juice, and $300ml$ water</span><br></br>
<span class="editorial">the second glass has $100$ml of lemon
juice, and $200$ml water</span></div>
<br></br>
<span class="editorial">What is the best way to compare these two
mixtures? There are several different ways, as shown by the
different explanations submitted</span>.
<p><span class="editorial">Mahir, from Saltus Grammar School
converted the values given so that there is the same amount of
water in each glass:</span></p>
<span class="editorial">In the first glass, there is $300$ml water,
and in the second, there is $200$ml. Therefore, to equate the
amounts of water, he multiplied everything in the first glass by
two, and everything in the second glass by three. Each glass now
has $600$ml water, and so can now be compared. Note that it is "ok"
to multiply</span> <span style="font-weight: bold; font-style: italic;" class="editorial">everything</span> <span class="editorial">in each
glass by a certain number as it is the</span> <span style="font-weight: bold; font-style: italic;" class="editorial">proportions</span> <span class="editorial">or</span> <span style="font-weight: bold; font-style: italic;" class="editorial">relative</span> <span class="editorial">amounts
we are interested in, rather than the absolute amounts.</span>
<p class="editorial">For the first glass, we now have:</p>
<p style="margin-left: 40px;" class="editorial">$400$ml lemon juice
and $600$ml water</p>
<p class="editorial">For the second glass, we now have:</p>
<p style="margin-left: 40px;" class="editorial">$300$ml lemon juice
and $600$ml water</p>
<p class="editorial">From this, we can now tell that the mixture in
the first glass must taste stronger: for the same amount of water,
there is more lemon juice.</p>
<p class="editorial">Jonathan, from Wilson's School, used a similar
method. However, he instead made the amounts of lemon juice equal,
and then saw which glass had more water. The glass with less water
for the same amount of juice will be stronger, as the juice is less
diluted.</p>
<p class="editorial">Another related method, used by many people
was to use ratios, fractions and/or percentages. Will K. from
Wilson's School gave a lovely explanation:</p>
Glass 1 would be stronger as it has a simplified ratio of lemon
juice: water of $2:3$, and so is $\frac{2}{5}$ lemon juice, or
$40$%, whereas glass $2$ has a simplified ratio of $1:2$ for lemon
juice: water, and so is $\frac{1}{3}$ lemon juice, or $33.\dot{3}$
% lemon juice, and so is weaker than glass $1$.<br></br>
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The strategy is to work out the ratio (e.g. $40$ml lemon juice to
$120$ml water would be $40:120$ lemon juice: water), simplify it
(e.g. $40:120$ simplifies to $4:12$, which simplifies to $1:3$),
and then turn that into a fraction ($1:3$ would be $\frac{1}{4}$ as
$1 +3 = 4$ and so the $1$ part that is lemon juice is $\frac{1}{4}$
of the drink), and then turn that into a percentage ($\frac{1}{4}$
into $25$%). The glass with the highest percentage being lemon
juice (or the lowest percentage being water) would be the strongest
glass of lemonade.<br></br>
<br></br>
<p><span class="editorial">Will expressed the strength of the
lemonade as a percentage, as these can be easily compared.
Sharumilan, also from Wilson's School converted the fractions so
that they had a common denominator. In this way, they can be more
easily compared. In fact, this is the same as converting to
percentages; percentages are fractions with a denominator of $100$!
Here is Sharumilan's answer:</span></p>
The first glass has the stronger tasting lemonade because
$\frac{2}{5}$ ($\frac{6}{15}$) of it is lemon juice while only
$\frac{1}{3}$ ($\frac{5}{15}$) of the lemonade in the second glass
is lemon juice.<br></br>
<br></br>
<p><span class="editorial">What about a more visual approach? Iona,
from Whitby Maths Club compared the two solutions by drawing</span>
<a href="/content/id/6870/Lemonade%20graph.pdf" class="editorial">these graphs</a><span class="editorial">.
Dominic, from Wilson's School, also suggested a graphical
approach:</span></p>
If you wanted to set it out in a pie chart you could easily see
which mixture was sronger. The graph with the biggest chunk of
lemon in it would be the strongest.
<p class="editorial">Another suggestion made by several people
would be to convert the amounts so that there are equivalent
amounts of water or lemon juice, as described above. Then, you
could draw a glass and visually see which is the stronger
mixture.</p>
<p class="editorial">Dulan, from Wilson's School, suggested a very
nice method, which can display graphically multiple different
mixtures:<br></br>
He thought that a graph could be constructed with $x$ and $y$
axes. On the $x$ axis could be "amount of lemon juice", and on the
$y$ axis, "amount of water". Try constructing this for
yourself.<br></br>
What does it mean if two mixtures have the same $x$ coordinate, but
different $y$ coordinates?<br></br>
What if they have the same $y$ coordinates, but different $x$
coordinates?<br></br>
You should be able to construct straight lines from the origin to
the various points representing different mixtures and compare
their strengths.<br></br>
Along each line the strength of all of the mixtures is the same as
the proportions do not change.</p>
<p class="editorial">We have now seen different examples of
approaches to this problem. Do the different methods always work?
Which method is most efficient?</p>
<p class="editorial">Nathan, from Wilson's School noted that
different methods are more efficient in different situations:</p>
The ratio or fraction strategy would be more efficient for
difficult fractions e.g. $\frac{300}{430}$ and $\frac{290}{560}$
but using a mental method would be more efficient for numbers like
$\frac{30}{50}$ and $\frac{40}{50}$, as you can see that
$\frac{40}{50}$ has a more concentrated supply of lemonade.<br></br>
<p class="editorial">Several people had their own preference of
method, depending on what they felt most comfortable using.</p>
<p class="editorial">Sharumilan explained this, and also examined
the combination of the different mixtures:</p>
I always use fractions because it seems easier for me and the graph
helped me to get the fractions quickly so it could actually be a
mix of both.<br></br>
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The strength of combined mixtures always has to be the same as or
in between the two mixtures. I have said "the same as" because if
the two mixtures are identical, the combined mixture has to be the
same as well. <br></br>
<p class="editorial">Try this out for yourself, with some squash,
or cordial...</p>
<p class="editorial">Thank you very much to everyone who submitted
solutions. There were many correct solutions, and so we could not
mention them all. Well done!</p>
<p><span class="editorial">If you enjoyed this problem, try the
extension problem</span> <a href="http://nrich.maths.org/6882&part=Ratios%20and%20Dilutions" class="editorial">Ratios and Dilutions</a><span class="editorial">.</span></p>
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<h3>Why do this problem?</h3>
<div><a href="http://nrich.maths.org/6870">This problem</a> gives a clear context in which fractions, ratio and proportion can be investigated. When using the interactivity, students can develop strategies for comparing fractions or ratios while thinking about which strategies are most useful for different cases.<br></br>
</div>
<h3>Possible approach</h3>
<div>Start by showing the class the image of the two glasses of lemonade. Ask them to decide which would taste stronger, and then share different ways of working it out, collecting different representations on the board. Then use the interactivity to generate a couple more examples for the class to discuss in the same way.</div>
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<div>If students have access to computers, set them to work in pairs using the interactivity, and challenge them to get ten correct answers in a row. A possible approach, if computers are unavailable, is to generate some questions and write them up on the board for students to work on with their partner. Alternatively, you could use the examples in <a href="/content/id/6870/Mixing%20Lemonade.doc">this worksheet</a>.</div>
<div> </div>
<p>Once students have had a chance to develop clear strategies for working out which mixtures taste strongest, share with students the questions to consider from the problem and allow them some time to explore and discuss their answers.<br></br>
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<em>You can read about <a href="https://www.ncetm.org.uk/resources/34329">one teacher's experience</a> of using this task in the classroom.</em></p>
<h3>Key questions</h3>
<div>Is there a strategy that always works?</div>
<div>Is your strategy always the quickest way to work out which mixture tastes strongest?<br></br>
</div>
<h3>Possible extension</h3>
<div>The mediant of two fractions $\frac{a}{b}$ and $\frac{c}{d}$ is found by adding the numerators and the denominators: $\frac{a+c}{b+d}$</div>
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<div>How could the lemonade problem be used to convince someone that the mediant is always in between the two original fractions?</div>
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<div><a href="http://nrich.maths.org/6882&part=">Ratios and Dilutions</a> looks at solution strength at a more challenging level.</div>
<h3>Possible support</h3>
<p>Spend plenty of time sharing alternative strategies for the first few simple examples on the worksheet, ensuring students understand <span style="font-weight: bold;">why</span> they work. Encourage students to work together using each other's methods to solve the harder examples.<br></br>
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<mdoxml version="1.0">One way of representing these beakers of juice without drawing the
beakers each time is to use fractions: $\frac{2}{5}$ of the first
beaker is lemon juice. $\frac{1}{3}$ of the second beaker is lemon
juice.<br></br>
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