$\frac{5}{9}(F - 32)$ The equation above shows how temperature $F$ , measured in degrees Fahrenheit, relates to a temperature $C$ , measured in degrees Celsius. Based on the equation, which of the following must be true? $\newline$ A temperature increase of $1$ degree Fahrenheit is equivalent to a temperature increase of $\frac{5}{9}$ degree Celsius. $\newline$ A temperature increase of $1$ degree Celsius is equivalent to a temperature increase of $1.8$ degrees Fahrenheit. $\newline$ A temperature increase of $\frac{5}{9}$ degree Fahrenheit is equivalent to a temperature increase of $1$ degree Celsius.

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Grade 8

Convert between Celsius and Fahrenheit

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\frac{5}{9}(F - 32)

The equation above shows how temperature

F

, measured in degrees Fahrenheit, relates to a temperature

C

, measured in degrees Celsius. Based on the equation, which of the following must be true?

\newline

A temperature increase of

1

degree Fahrenheit is equivalent to a temperature increase of

\frac{5}{9}

degree Celsius.

\newline

A temperature increase of

1

degree Celsius is equivalent to a temperature increase of

1.8

degrees Fahrenheit.

\newline

A temperature increase of

\frac{5}{9}

degree Fahrenheit is equivalent to a temperature increase of

1

degree Celsius.

Given Equation: We are given the equation that relates temperature in Fahrenheit (F) to temperature in Celsius (C): $\newline$ $C = \frac{5}{9}(F - 32)$ $\newline$ To find the relationship between a temperature increase in Fahrenheit and a temperature increase in Celsius, we need to consider how a change in Fahrenheit affects Celsius.
Temperature Increase Analysis: Let's consider an increase of $1$ degree Fahrenheit. We can represent this as $F + 1$ . To find out how much this increases the temperature in Celsius, we substitute $F + 1$ into the equation in place of $F$ : $\newline$ $C_{new} = \frac{5}{9}((F + 1) - 32)$
Simplifying Equation: We can simplify the equation by distributing the $\frac{5}{9}$ across the terms inside the parentheses: $\newline$ $C_{new} = \frac{5}{9}F + \frac{5}{9} - \frac{5}{9} \times 32$
Temperature Increase in Celsius: The original temperature in Celsius is given by: $\newline$ $C = \frac{5}{9}(F - 32)$ $\newline$ So the increase in temperature in Celsius ( $\Delta C$ ) when Fahrenheit increases by $1$ degree is: $\newline$ $\Delta C = C_{new} - C$ $\newline$ $\Delta C = \left(\frac{5}{9}F + \frac{5}{9} - \frac{5}{9} \times 32\right) - \left(\frac{5}{9}F - \frac{5}{9} \times 32\right)$
Reverse Analysis: Notice that the terms $\frac{5}{9}F$ and $-\frac{5}{9} \times 32$ will cancel out because they are present in both $C_{new}$ and $C$ , leaving us with: $\newline$ $\Delta C = \frac{5}{9}$ $\newline$ This means that an increase of $1$ degree Fahrenheit is equivalent to an increase of $\frac{5}{9}$ degree Celsius.
Isolating New Fahrenheit: Now let's consider the reverse: an increase of $1$ degree Celsius. We want to find out how much this increases the temperature in Fahrenheit. We can start by adding $1$ to the Celsius temperature and then solving for the new Fahrenheit temperature. $\newline$ $C + 1 = \frac{5}{9}(F_{new} - 32)$
Temperature Increase in Fahrenheit: To isolate $F_{new}$ , we multiply both sides by $\frac{9}{5}$ and then add $32$ : $\newline$ $\frac{9}{5}(C + 1) = F_{new} - 32$ $\newline$ $F_{new} = \frac{9}{5}(C + 1) + 32$
Temperature Increase in Fahrenheit: To isolate $F_{new}$ , we multiply both sides by $\frac{9}{5}$ and then add $32$ : $\newline$ $\frac{9}{5}(C + 1) = F_{new} - 32$ $\newline$ $F_{new} = \frac{9}{5}(C + 1) + 32$ The original Fahrenheit temperature is given by: $\newline$ $F = \frac{9}{5}C + 32$ $\newline$ So the increase in temperature in Fahrenheit ( $\Delta F$ ) when Celsius increases by $1$ degree is: $\newline$ $\Delta F = F_{new} - F$ $\newline$ $\Delta F = \left(\frac{9}{5}(C + 1) + 32\right) - \left(\frac{9}{5}C + 32\right)$
Temperature Increase in Fahrenheit: To isolate $F_{new}$ , we multiply both sides by $\frac{9}{5}$ and then add $32$ : $\newline$ $\frac{9}{5}(C + 1) = F_{new} - 32$ $\newline$ $F_{new} = \frac{9}{5}(C + 1) + 32$ The original Fahrenheit temperature is given by: $\newline$ $F = \frac{9}{5}C + 32$ $\newline$ So the increase in temperature in Fahrenheit ( $\Delta F$ ) when Celsius increases by $1$ degree is: $\newline$ $\Delta F = F_{new} - F$ $\newline$ $\Delta F = \left(\frac{9}{5}(C + 1) + 32\right) - \left(\frac{9}{5}C + 32\right)$ Again, the terms $\frac{9}{5}C$ and $32$ will cancel out, leaving us with: $\newline$ $\Delta F = \frac{9}{5}$ $\newline$ This means that an increase of $1$ degree Celsius is equivalent to an increase of $\frac{9}{5}$ degrees Fahrenheit, which is the same as $1$ . $8$ degrees Fahrenheit.