Please help me understand Pizza + Donut logic

I’m almost 50, and to me, mathematics has always been contrived number games based on a deeper logic always beyond my grasp. Yet through my interests in physics, IT, music and astronomy, I know that mathematics is an amazingly adaptable language which can be used to express (and therefore model) any process in the universe one cares to apply it to.

The “Pizza + Donut” quiz in Basic Mathematics is a perfect example of the mathematical logic which has confounded me since childhood. I fervently wish, with all my heart, that I could follow the logic in the statement: “If a pizza costs $10.50 more than a donut, then a pizza and a donut must cost $10.50 more than two donuts.” I absolutely cannot fathom the logic behind that statement. Why two donuts? If we don’t know the cost of one donut, how can we possibly say the difference between a pizza and donut combined is 10.50 more than two donuts?

Please, please, can someone help me follow that logic? I believe if I could only grasp it, a new understanding of this universe would be unlocked for me.

Note by Alan Mellerick
6 months, 4 weeks ago

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Let \(P \) and \(D\) denote the price of a pizza and a donut (in $), respectively.

The first part of the sentence says that \(P = 10.50 + D\).

Now, if we add \(D\) to both sides of this equation, we will get \(P + D = 10.50 + 2D\).

What does \(P+D\) and \(10.50 + 2D\) represents?

Pi Han Goh - 6 months, 4 weeks ago

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OK, I see how you can turn the statement into an equation - and it does make it easier to think about! So now I can see why you might put two donuts on the right side of the equation if you add one to the left. Great - progress! But in trying to solve for D, I just can’t reason it out; I don’t know what it represents.

Alan Mellerick - 6 months, 4 weeks ago

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You can't solve for \(D\) because you are only given 1 (linear) equation with 2 variables.

Pi Han Goh - 6 months, 4 weeks ago

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@Pi Han Goh OK - so... Does that mean the original question posed is somehow flawed? This all comes from the “Doubtful Discounts” exercise on the Maths Fundamentals on Brilliant. I was struggling through the explanation given for the correct answer. Perhaps they presented the statement “...if a pizza costs $10.50 more than a donut, then a pizza and a donut cost $10.50 more than 2 donuts.” as a step to understanding the answer. I can at least follow the logic in that sentance now - thank you very much for your assistance with that. I still don’t understand why one would go down that path as a technique in solving the the donut price. I’ll try harder!

Alan Mellerick - 6 months, 4 weeks ago

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@Alan Mellerick

Does that mean the original question posed is somehow flawed?

No. Just because you don't have enough information to solve an equation (or multiple equation) doesn't automatically makes the question flawed.

I still don’t understand why one would go down that path as a technique in solving the the donut price. I’ll try harder!

Try more problems. You'll get a hang of it! ;)

Pi Han Goh - 6 months, 4 weeks ago

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Wow. That is one expensive donut

Annie Li - 6 months, 4 weeks ago

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