Question

If the temperature goes up \(1 \frac{1}{4}^{\circ} \mathrm{F}\), it will be \(0^{\circ} \mathrm{F}\). What is the temperature now?

Short answer

-1\(^{\circ} \mathrm{F}\) is the temperature now.

Step-by-step solution

Understand the Problem

We need to find the current temperature given that increasing this temperature by \(1 \frac{1}{4}^{\circ} \mathrm{F}\) results in \(0^{\circ} \mathrm{F}\).

Convert the Mixed Number

Convert the mixed number \(1 \frac{1}{4}\) to an improper fraction. \(1 \frac{1}{4} = 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}\).

Set Up the Equation

Let the current temperature be \(T\). When the temperature increases by \( \frac{5}{4}^{\circ} \mathrm{F} \), it becomes \(0^{\circ} \mathrm{F}\). Thus, we set up the equation: \(T + \frac{5}{4} = 0\).

Solve the Equation

To find \(T\), isolate \(T\) on one side of the equation.\(T = 0 - \frac{5}{4} = - \frac{5}{4}\).

Convert Improper Fraction to Mixed Number

Convert \(- \frac{5}{4}\) back to a mixed number for easier interpretation. \(- \frac{5}{4} = - 1 \frac{1}{4}\).

State the Answer

The current temperature is \(- 1 \frac{1}{4}^{\circ} \mathrm{F}\).

Key concepts

temperature conversion

Understanding temperature conversions is crucial in many mathematical problems. In this exercise, we are dealing with Fahrenheit temperatures. To solve the problem, we need to increase the current temperature by a certain amount and reach a final value.

To solve temperature problems, we often perform these steps:

• Identify the starting temperature.
• Identify the change or difference in temperature.
• Add or subtract the temperature difference to find the final temperature.

In our specific exercise, the given change is an increase by \(1 \frac{1}{4}^{\circ} \mathrm{F}\), resulting in a temperature of \(0^{\circ} \mathrm{F}\). To find the starting temperature, we need to set up and solve an equation, which we will delve into in the sections below.

mixed numbers

A mixed number is a combination of a whole number and a fraction, such as \(1 \frac{1}{4}\). When working with mixed numbers in mathematical equations, it’s often easier to first convert them into improper fractions.

Here’s how to convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.
• Place the total over the original denominator to form an improper fraction.

For our problem, we converted \(1 \frac{1}{4}\) to an improper fraction:
\(1 \cdot 4 + 1 = 5\) and thus \(1 \frac{1}{4} = \frac{5}{4}\).
This improper fraction makes it easier to set up and solve our temperature equation.

solving equations

Solving equations is a critical mathematical skill. In this exercise, we need to solve for the unknown starting temperature. First, we set up an equation based on the problem statement: let the current temperature be \(T\). Increasing \(T\) by \(\frac{5}{4}^{\circ} \mathrm{F}\) results in \(0^{\circ} \mathrm{F}\). Thus, we write:

\(T + \frac{5}{4} = 0\)

To isolate \(T\), we subtract \(\frac{5}{4}\) from both sides:

\(T = 0 - \frac{5}{4} \)
\(T = -\frac{5}{4} \).

Finally, converting \(-\frac{5}{4}\) back to a mixed number for easier interpretation gives us \(-1 \frac{1}{4}\).

Hence, the current temperature is \(–1 \frac{1}{4}^{\circ} \mathrm{F}\). Breaking down equations into smaller steps and converting numbers carefully ensures accurate results.