Subsection 1.1.6 Real-world applications of reciprocals of real numbers
Reciprocals are important in various fields, including finance, science, engineering, medicine, and transportation . In disciplines such as physics, chemistry, and manufacturing, dividing rational numbers using reciprocals helps solve real-world problems, from calculating speed and time to determining chemical concentrations and optimizing production efficiency.
In this section, we will explore real-world applications of reciprocals and how they make mathematical computations easier and more practical.
Activity 1.1.8.
Nkirote is a baker and is preparing large orders of bread for a grade
\(10\) student’s event. Each batch of bread requires
\(\frac{2}{3}\) kg of flour, and she has a total of
\(12\) kg of flour. She needs to determine how many full batches she can make. As she prepares, she realizes that each batch also requires
\(\frac{3}{4}\) liters of milk. If she has
\(9\) liters of milk available, she must check whether she has enough for all the batches. To ensure efficiency, she calculates how many full loaves of bread she can produce if each batch yields
\(\frac{2}{5}\) of a loaf per kilogram of flour used.
1. Working in groups, use reciprocals to determine how many full batches of bread the baker can make with
\(12\) kg of flour.
2. Calculate whether
\(9\) liters of milk is enough for all the batches.
3. If each batch produces
\(\frac{2}{5}\) of a loaf per kilogram of flour, how many full loaves can she make?
4. How was reciprocal helpful in dividing quntities while baking?
5. Discuss areas where reciprocals are applied in the real word.
6. Discuss your work with fellow learners.
\(\textbf{Key Takeaway}\)
Here are practical examples of how reciprocals are applied to solve real-world problems:
1. A printing machine can print
\(\frac{5}{6}\) of a book page per second. How long will it take to print
\(20\) pages?
If
\(1\) second =
\(\frac{5}{6}\) of a page, then for
\(20\) pages, we calculate as follows:
\begin{align*}
\text{ Time } =\amp 20 \div \frac{5}{6}\\
=\amp 20 \times \frac{6}{5}
\end{align*}
Multiplying and simplifying we have: Time =
\(\frac{120}{5} = 24\) seconds.
Thus, the printing machines will take
\(24\) seconds to print
\(20\) pages.
2. If a car travels at
\(80\) km/h, to find out the time taken per km we can use reciprocal.
Time per km =
\(\frac{1}{80}\) hours per km.
3. If a factory produces
\(300\) items in
\(5\) hours, then the production rate per hour is:
\(\frac{300}{5} = 60\) items per hour.
To determine how much time is needed per item, we use reciprocal:
Hence, the factory produces
\(1\) item every
\(\frac{1}{60}\) hours.
Example 1.1.24.
Okoth runs
\(3 \frac{1}{4}\) miles in
\(\frac{1}{2}\) an hour. What is her speed in miles per hour?
Solution.
To find speed we divide distance by time such that: speed =
\(\frac{\text{ distance}}{\text{ time}}\text{.}\)
First, we need to convert the fraction
\(3\frac{1}{4}\) into an improper fraction which is
\(\frac{13}{4}\)
Hence, speed =
\(3\frac{1}{4} \div \frac{1}{2} = \frac{13}{4} \div \frac{1}{2}\)
Multiply
\(\frac{13}{4}\) by the reciprocal of
\(\frac{1}{2}\text{.}\)
\(\frac{13}{4} \times \frac{2}{1} = \frac{26}{4}\)
Simplifying
\(\frac{26}{4}\) we get
\(\frac{13}{2}\) which is
\(= 6 \frac{1}{2}\text{.}\)
1. A paint manufacturer claims that
\(1\) liter of paint covers
\(48\) square feet. A customer buys a
\(0.5\)-liter sample and uses it to paint a
\(2\)-foot by
\(4\)-foot section of a wall. To determine whether the sample meets the manufacturer’s claim, first calculate the area it covers. Then, using reciprocals, find how many times
\(0.5\) liters fits into
\(1\) liter. Compare the result with the stated
\(48\) square feet per liter to determine if the sample performs as expected.
2. A gardener can plant
\(\frac{7}{9}\) of a flowerbed per hour. How long will it take to complete
\(18\) flowerbeds?
3. A printer in a publishing company prints
\(\frac{2}{5}\)of a book page per second. How long will it take to print
\(90\) pages at this rate?
4. A car consumes
\(\frac{2}{7}\)of a gallon of fuel per mile. If the fuel tank holds
\(21\) gallons, how many miles can the car travel before running out of fuel?
5. A construction crew is tasked with laying concrete on a road section. A
\(7\) -ton cement truck can produce enough concrete to cover
\(280\) square meters. However, due to supply limitations, only
\(4.5\) tons of cement is available. Using reciprocals, determine how many square meters can be covered with
\(4.5\) tons of cement.
6. A laboratory is preparing a specialized chemical solution. The standard formula states that
\(5 \) liters of a concentrated acid can be diluted to produce
\(150\) liters of usable solution. If the lab technician only has
\(3.75\) liters of the concentrate, use reciprocals to calculate how many liters of usable solution can be prepared.
7. A group of workers can complete
\(\frac{3}{4}\) of a construction task in
\(6\) hours. How much time will it take to finish the entire task at the same rate?
