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2023 AMC 10A Problem 8

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2023 AMC 10A Problem 8AMC 10 Step by Step
Name:                             Date: September 13, 2026
Linear Equations & Word Problems · difficulty 2 of 5 · about 2 min · position P1-10

Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at 110110 degrees Fahrenheit, which is 00 degrees on the Breadus scale. Bread is baked at 350350 degrees Fahrenheit, which is 100100 degrees on the Breadus scale. Bread is done when its internal temperature is 200200 degrees Fahrenheit. What is this, in degrees, on the Breadus scale?

  1. A)

    3333

  2. B)

    34.534.5

  3. C)

    3636

  4. D)

    37.537.5

  5. E)

    3939

Answer

D

Key insight

A linear scale preserves proportions: 200 is 90/240 = 3/8 of the way from 110 to 350, so it is 3/8 of the way from 0 to 100.

Solution

The two scales are related linearly, so equal fractions of the corresponding intervals match. In Fahrenheit, the interval from 110110 to 350350 has length 240240, and 200200 sits
200−110240=90240=38\frac{200 - 110}{240} = \frac{90}{240} = \frac{3}{8}
of the way along it. On the Breadus scale the same interval runs from 00 to 100100, so the reading is 38⋅100=37.5\frac38 \cdot 100 = 37.5.

Equivalently, the slope is 100240=512\frac{100}{240} = \frac{5}{12} Breadus degrees per Fahrenheit degree, and B=512(F−110)B = \frac{5}{12}(F - 110) gives B=512⋅90=37.5B = \frac{5}{12} \cdot 90 = 37.5.

The answer is (D) 37.5\boxed{\textbf{(D)}\ 37.5}.

Why this works

Two points determine a linear function, and the cleanest way to use them is "fraction of the way along the interval," which is the same on both scales. Writing B=m(F−110)B = m(F - 110) with the known zero point avoids solving for an intercept. This is exactly how Celsius and Fahrenheit conversions are derived.

The trap

Using the ratio 200/350 of raw temperatures instead of the fractions of the intervals measured from the matching endpoints.

Common mistakes

  • Using the ratio 200/350 of raw temperatures instead of the fractions of the intervals measured from the matching endpoints.
  • Computing the slope as 240100\frac{240}{100} (Fahrenheit per Breadus) and multiplying instead of dividing.

Techniques

Set up the equation/formula and compute; no special trick needed