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Algebra Difficulty 3.1 AMC 10/12 Prove it Canada

With 1 kg of muffin batter, exactly $24$ mini muffins and $2$ large muffins can be made. With 2 kg of muffin batter, exactly $36$ mini muffins and $6$ large muffins can be made. With 2 kg of muffin batter, exactly $48$ mini muffins and $n$ large muffins can also be made. What is the value of $n$ ? With $x$ kg of muffin batter, exactly 84 mini muffins and 10 large muffins can be made. What is the value of $x$ ?
Determine how many mini muffins can be
made using the same amount of batter that is needed to make 7 large
muffins.

Solution

With 1 kg of muffin batter, exactly 24 mini muffins and 2 large
muffins can be made. Thus with 2 kg of muffin batter, exactly $2\times24=48$ mini muffins and $2\times2=4$ large muffins can be made, and so $n=4$ . Solution 1 With 2 kg of muffin batter, exactly 36 mini muffins and 6 large muffins can be made. With 2 kg of muffin batter, exactly 48 mini muffins and 4 large muffins can be made. Adding these, we get that with $2 $\textrm{}$ kg}+2 $\textrm{}$ kg}= 4 $\textrm{}$
kg} $of muffin batter, exactly$ 36+48=84 $mini muffins and$ 6+4=10 $large muffins can be made, and so$ x=4 $. Solution 2 Let$ m be the number of kilograms of muffin batter needed to make 1 mini muffin. Let

Figure for this problem

Figure for this problem $\ell$ be the number of kilograms of muffin batter needed to make 1 large muffin. Since 1 kg of muffin batter makes exactly 24 mini muffins and 2 large muffins, then

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Figure for this problem $1=24m+2\ell$ . Since 2 kg of muffin batter makes exactly 36 mini muffins and 6 large muffins, then

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Figure for this problem $2=36m+6\ell$ . Subtracting the second equation from 3 times the first equation, we get

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Figure for this problem $3\times1-2=3\times(24m+2\ell)-(36m+6\ell)$ $or$ 1=36m $, and so$ $m=\dfrac{1}{36}$ $. Substituting$ $m=\dfrac{1}{36}$ into the second equation, we get

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Figure for this problem $2=36\left(\dfrac{1}{36}\right)+6\ell$ $or$ $1=6\ell$ $, and so$ $\ell=\dfrac16$ $. Since$ $\dfrac{1}{36}$ kg of muffin batter is needed to make 1 mini muffin, then

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Figure for this problem $84\cdot\dfrac{1}{36}=\dfrac73$ kg of muffin batter is needed to make 84 mini muffins. Since

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Figure for this problem $\dfrac{1}{6}$ kg of muffin batter is needed to make 1 large muffin, then

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Figure for this problem $10\cdot\dfrac{1}{6}=\dfrac53$ kg of muffin batter is needed to make 10 large muffins. Thus, exactly 84 mini muffins and 10 large muffins can be made with

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Figure for this problem $\dfrac73+\dfrac53=\dfrac{12}{3}=4$ $kg of muffin batter, and so$ x=4 $. In part (b) Solution 2, it was determined that$ $\frac{1}{36}$ kg of muffin batter is needed to make 1 mini muffin, and

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Figure for this problem $\frac{1}{6}$ kg of muffin batter is needed to make 1 large muffin. Therefore, the number of kilograms of muffin batter needed to make 1 large muffin is 6 times the number of kilograms of batter needed to make 1 mini muffin

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Figure for this problem $\left(6\times\frac{1}{36}=\frac16\right)$ . In other words, the batter needed to make 1 large muffin can make 6 mini muffins. Then, using the amount of batter that makes 7 large muffins,

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Figure for this problem $6\times7=42\$$ mini muffins can be
made.

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