Exploration 1.4.1.
An example of a decimal number is \(3.75\text{.}\) The whole number part is 3,the decimal part is 0.75 which represents 75 hundredths.
Section 1.4 Decimals
Murunga a grade \(7\) student carries a bag weighing \(450 g\text{.}\) He can convert the mass of his bag to kg i.e \(\frac{450 g}{1000 g} \times 1 kg = 0.45 kg\text{.}\) Murunga used decimals to identify the mass of his bag in kg.
Subsection 1.4.1 Place value of digits in decimals
Place value of digits in a decimal number refers to the position of each digit in a decimal number, which determines its value. As we move to the right of the decimal point, each place value is 10 times smaller than the one before it.
Activity 1.4.2.
1. Working in groups of five, write down each odd number from \(0-9\) on a different piece of paper.
2. Draw a place value chart similar to the one below.
| Number | Ones | Decimal point | Tenths | Hundredths | Thousandths | Ten thousandths |
|---|---|---|---|---|---|---|
| \(\) | \(\) | \(.\) | \(\) | \(\) | \(\) | \(\) |
| \(\) | \(\) | \(.\) | \(\) | \(\) | \(\) | \(\) |
| \(\) | \(\) | \(.\) | \(\) | \(\) | \(\) | \(\) |
| \(\) | \(\) | \(.\) | \(\) | \(\) | \(\) | \(\) |
| \(\) | \(\) | \(.\) | \(\) | \(\) | \(\) | \(\) |
3. Shuffle the pieces of paper and in turns, each learner in the group to pick a piece of paper and form a number using the digits on the pieces of paper.
4. Divide the number formed by the place value of the first digit in the that number. Example: the number formed is \(35\,791\) then the place value of \(3\) is tens of thousands hence \(\frac{35 791}{10 000} = 3.5791\text{.}\)
5. Write the decimal number in the place value chart as shown below and identify place values of each digit.
| Number | Ones | Decimal point | Tenths | Hundredths | Thousandths | Ten thousandths |
|---|---|---|---|---|---|---|
| \(4.6852\) | \(4\) | \(.\) | \(6\) | \(8\) | \(5\) | \(2\) |
| \(\) | \(\) | \(.\) | \(\) | \(\) | \(\) | \(\) |
| \(\) | \(\) | \(.\) | \(\) | \(\) | \(\) | \(\) |
| \(\) | \(\) | \(.\) | \(\) | \(\) | \(\) | \(\) |
| \(\) | \(\) | \(.\) | \(\) | \(\) | \(\) | \(\) |
6. Discuss and compare your results with fellow learners.
Example 1.4.1.
Use a place value chart to identify and write the place value of each digit in the following numbers.
Solution.
Draw a place value chart and represent the numbers as below.
| Number | Hundreds | Tens | Ones | Decimal point | Tenths | Hundredths | Thousandths | Ten thousandths | Hundred thousandths |
|---|---|---|---|---|---|---|---|---|---|
| \(\text{a) } 138.45239\) | \(1\) | \(3\) | \(8\) | \(.\) | \(4\) | \(5\) | \(2\) | \(3\) | \(9\) |
| \(\text{b) } 92.35798 \) | \(9\) | \(2\) | \(.\) | \(3\) | \(5\) | \(7\) | \(9\) | \(8\) | |
| \(\text{c) } 432.21648\) | \(4\) | \(3\) | \(2\) | \(.\) | \(2\) | \(1\) | \(6\) | \(4\) | \(8\) |
Checkpoint 1.4.2.
Subsection 1.4.2 Total value of digits in decimals
The total value of digits in a decimal is found by multiplying each digit by its place value and summing the results.
The place value decreases in powers of 10 as you move right of the decimal point.
For example, in \(2.45\text{,}\) the total value is: each digit multiplied by its place value as below
\begin{align*}
2.45 \amp= (2 \times 1) + (4 \times 0.1) + (5 \times 0.01) \\
\amp= 2 + 0.4 + 0.05
\end{align*}
\(\textbf{Explore}\)
In a school race , Tula run 2.45 km—that is 2 full kilometers, 400 meters (4 tenths of a km), and 50 meters (5 hundredths of a km). We use total value of digits in a decimal to identify the distance Talu ran that could not form a whole number.
Activity 1.4.3.
1. Working in groups, think of a number that has \(5\) decimal points i.e \(18.52847,7.24385\) and in turns write it down.
2. Discuss the chosen decimal numbers with fellow group members.
3. Draw a place value chart and write the decimal numbers on the chart.
4. Identify the place value of each digit on the chart then identify the total value of each digits.
5. With your fellow learners, discuss how to identify the total value of digits in a decimal number.
Example 1.4.3.
What is the total value of each digit in the number \(35.58942\)
Solution.
| Number | PLace value | Total value |
|---|---|---|
| \(3\) | Tens | \(3 \times 10 = 30\) |
| \(5\) | Ones | \(5 \times 1 = 5\) |
| . | ||
| \(5\) | Tenths | \(5 \times 0.1 = 0.5\) |
| \(8\) | Hundredths | \(8 \times 0.01 = 0.08\) |
| \(9\) | Thousandths | \(9 \times 0.001 = 0.009\) |
| \(4\) | Ten thousandths | \(4 \times 0.0001 = 0.0004\) |
| \(2\) | Hundred thousandths | \(2 \times 0.00001 = 0.00002\) |
Checkpoint 1.4.4.
Subsection 1.4.3 Multiplying decimals by a whole number
Akinyi has a bottle of juice she prepared for her friends. She pours 0.4 liters of juice into each of the \(5\) glasses she had and realized all the glasses was filled without any juice remaining in the bottle. We can find out the total amount of juice Akinyi had by applying multiplication of decimals. To find how much juice Akinyi had we multiply the amount amount of juice in each glass with the number of glasses i.e \(0.4 \times 5 = 2\)
Activity 1.4.4.
1. Measure the length and width of your classroom in cm using a meter ruler and write it down.
2. Discuss the measurement of one side of your classroom and convert the measurement to \(m\) i.e \(1 m = 100 cm \text{.}\)
3. Multiply the length and width in \(m\) by the number of sides in your classroom. What do you get?
4. How many classrooms do you have in your school?
5. Suppose all the classrooms has same lengths and widths, determine the total measurement of both length and width of classrooms in your school.
6. How did you determine the total measurement of length and width of classrooms in the school?
Example 1.4.6.
Work out:
Solution.
-
\(\displaystyle 8.653 \times 10\)
-
\(\displaystyle 8.653 \times 100\)
Example 1.4.7.
Solution.
| \(19342 \) | ||
| \(\times \, 21 \) | ||
| \(19342\) | \(\to\) | \(19342 \times 1\) |
| \(+\,386840\) | \(\to\) | \(19342 \times 2\) |
| \(406182\) |
-
Identify the number of decimal places i.e
-
Express 19.342 as a fraction = \(\frac{19342}{1000}\)
-
In the product, count \(3\) digits from the right towards the left and insert the decimal point. This is because the result of the product is supposed to be divided by \(1000\text{.}\)
\(\qquad \qquad \quad = 406.182\)
\(19.342 \times 21 = 406.182\)
Checkpoint 1.4.8.
Subsection 1.4.4 Multiplying decimals by decimals
Ekadeli is baking a cake. The recipe says that for every 1 kg of flour, she needs 0.4 kg of sugar. But today, she is using 2.5 kg of flour. For Ekadeli to identify how much sugar she will use she will apply multiplication of decimal number by a decimal number. Therefore, Ekadeli will multiply the amount of flour she is using by the amount of sugar used per kg. i.e
\(2.5 \times 0.4 = \frac{25 }{10 } \times \frac{4 }{10 } = \frac{100 }{100} = 1 kg\)
Activity 1.4.5.
Grade \(7\) pupils was given a square garden by the school to plant their vegetables and each side of the garden measured \(8.5 m\text{.}\)
1. Draw the shape of the vegetable garden and label its sides.
2. What is the area of the square garden in \(m\text{?}\) Remember area of a square = side \(\times\) side.
3. How did you multiplied the decimals? Share it with your fellow learners.
Example 1.4.9.
Find the area of the rectangle below.
Example 1.4.10.
Work out: \(7.215 \times 6.4\)
Solution.
| \(7\,215\) | ||
| \(\times \, 64\) | ||
| \(28\,860\) | \(\to\) | \(7\,215 \times 4\) |
| \(+\,43\,290\) | \(\to\) | \(7\,215 \times 6\) |
| \(461\,760\) |
\(= 46.1760\)
\(= 46.176\)
Checkpoint 1.4.11.
Subsection 1.4.5 Division of a decimal by a whole number
Moraa has 5.6 kg of sugar and wants to pack it equally into 4 small bags. Moraa can use division of decimal by a whole number to find out how many kilograms of sugar go into each bag.
Activity 1.4.6.
The figure below represents Kazani Primary School’s compound which is rectangular in shape.
1. What is the width of the compound if the length is \(7 m \) and the area is \(1.12 m^2\) ? (You may use a calculator to work out if you have or use the long division method)
2. Discuss how you can divide a decimal number by a whole number.
3. Share your work with other learners. 3. Share your work with other learners in the classroom .
Example 1.4.12.
Work out: \(3.75 \div 8\)
Solution.
| \(3.75 \div 8 = \frac{375}{100} \div 8\) | Here we make \(3.75\) a fraction. | |
| \(= \frac{375}{100} \times \frac{1}{8}\) | Then we multiply \(\frac{375}{100}\) by the reciprocal of \(\frac{8}{1}\) | |
| \(= \frac{\cancel{375}}{\cancel{800}} = \frac{15}{32}\) | Dividing \(\frac{375}{800}\) by \(25\text{,}\) it is then simplified to \(\frac{15}{32}\) | |
| \(= 0.46875\) | This is a result of converting \(\frac{15}{32}\) into a decimal | Writing \(0.46875\) into \(2\) decimal places gives \(0.47\) |
Checkpoint 1.4.13.
Subsection 1.4.6 Division of a decimal number by a decimal number
To divide decimal numbers you:
-
Express the division as a fraction .Example \(6.3 ÷ 0.7 \) can be written as\(\frac{63}{10} \div \frac{7}{10}\)
-
Convert division into multiplication that is instead of dividing, multiply the first fraction by the reciprocal of the second fraction i.e
Exploration 1.4.7.
In her daily life, Cherotich often measures milk and juice in liters. Today, she has a 6.3 liter container of milk and wants to pour it into small bottles, each holding 0.7 liters.Therefore, Cherotich uses division of decimal number by a decimal number to find out how many bottles she needs. Just like how we use whole numbers to count objects, Cherotich also uses division of decimals to measure and divide quantities accurately.
Activity 1.4.8.
\(\textbf{Instructions for teachers or team leaders}\)
1.Teacher or a team leader to write down on piece of paper or print the following questions:
-
12.4 ÷ 0.2 = ?(Find the next challenge around your classroom door.)
-
5.6 ÷ 0.7 = ?( The next station is at the corner of the classroom )
-
12.04 ÷ 1.3 = ?( Go to assembly ground near flagpole)
-
9.9 ÷ 3.3 = ?(Next station is near the school entrance )
-
18.75 ÷ 2.5 = ?( The final challenge is at the back of the class)
-
7.92 ÷ 1.2 = ?(Return to the teacher/ team leader for the last answer check)
2.Hide decimal division question papers around the specified areas on each questions.
\(\textbf{For students}\)
3. Working in groups of \(5\text{,}\) each group to choose a team captain responsible for reading questions and keeping the team organized.
4. Each group is given the first piece of paper containing questions and each paper has a problem and a clue leading to the next question.
\(\textbf{ Take one piece of paper after you find it and hide the rest around that place . This applies to all questions}\)
5. Each group to start at a different station (e.g., one in class, another at the assembly ground, another near the school entrance) and solves the problem on their question papers together. Correct answers earn the group the next clue to find the next question.
6. The first team to solve all questions correctly wins 🎁.
7.Each group to create their own decimal division problems for other groups to solve.
Example 1.4.14.
Erick had a sugarcane whose length was \(6.84\) m. He cut the sugarcane into equal pieces of \(2.28 \) m. How many pieces did he get?
Solution.
Equal pieces of sugarcane Erick got is given by:
\(6.84 \text{ m } \div 2.28 \text{ m } = \frac{6.84}{2.28}\)
Multiply both the numerator and the denomenator by \(100\) to eliminate the decimals.
\begin{align*}
\frac{6.84 \times 100}{2.28 \times 100} \amp = \frac{684}{228}\\
\amp \\
\amp = 3
\end{align*}
Therefore, Erick got \(3\) pieces of sugarcane.
Example 1.4.15.
Work out: \(9.45 \div 2.25 = \)
Solution.
Then multiply \(\frac{945}{100}\) by \(\frac{100}{225} \) which is the reciprocal of \(\frac{225}{100} \text{,}\) to get:
| \(21\) | \(1\) | ||
| \(\frac{\cancel{945}}{\cancel{100}}\) | \(\times\) | \(\frac{\cancel{100}}{\cancel{225}}\) | \(= \frac{21}{5}\) |
| \(1\) | \(5\) |
| \(4.2\) | |
| \(5\) | \(21\) |
| \(-20\) | |
| \(10\) | |
| \(-\,10\) | |
| \(00\) |
From the above, \(45\) is the common factor hence \(\frac{945}{225} \div \frac{45}{45}\) gives \(\frac{21}{5}\)
Therefore, \(9.45 \div 2.25 = 4.2 \)
