Rounding and Approximating Arithmetic Results for JSS 1

Approximation

Performance Objectives

By the end of the lesson, students should be able to:

1. approximate answers to addition and subtraction problems to a given degree of accuracy;
2. approximate answers to multiplication and division problems to a given degree of accuracy;
3. round numbers to the nearest 10, 100 and 1,000;
4. apply approximation involving basic operations in everyday life activities;
5. solve problems on quantitative aptitude involving approximation.

Meaning of Approximation

Approximation is the process of finding a value that is close enough to the correct value, but not necessarily exact.

In mathematics and everyday life, exact numbers are not always necessary. An approximate value is often easier to calculate, remember and use for quick decisions. For example, if a school has $597$ students, it is often simpler to say the school has about $600$ students.

Rounding Numbers to the Nearest 10, 100 and 1,000

Rounding is the most common method of approximation. To round a whole number to a given place value, the digit immediately to the right of that place value must be examined.

The standard rules for rounding are:

1. If the digit to the right is $5$ or more ($5, 6, 7, 8, 9$), round up. This means $1$ is added to the digit in the target place value, and all digits to its right become zeros.
2. If the digit to the right is less than $5$ ($0, 1, 2, 3, 4$), round down. This means the digit in the target place value remains the same, and all digits to its right become zeros.

Observe the number line diagram below, which shows how numbers are rounded to the nearest $10$. Numbers from $30$ to $34$ are closer to $30$, while numbers from $35$ to $40$ are closer to $40$.

Rounding to the Nearest 10

To round a number to the nearest $10$, the units digit must be examined.

The following examples show how to round numbers to the nearest $10$:

1. Round $73$ to the nearest $10$. The units digit is $3$. Since $3$ is less than $5$, the tens digit remains $7$ and the units digit becomes $0$. The result is $70$.
2. Round $286$ to the nearest $10$. The units digit is $6$. Since $6$ is $5$ or more, $1$ is added to the tens digit ($8 + 1 = 9$), and the units digit becomes $0$. The result is $290$.
3. Round $1,045$ to the nearest $10$. The units digit is $5$. Since $5$ is $5$ or more, $1$ is added to the tens digit ($4 + 1 = 5$), and the units digit becomes $0$. The result is $1,050$.

Rounding to the Nearest 100

To round a number to the nearest $100$, the tens digit must be examined.

The following examples show how to round numbers to the nearest $100$:

1. Round $438$ to the nearest $100$. The tens digit is $3$. Since $3$ is less than $5$, the hundreds digit remains $4$, and the tens and units digits become $0$. The result is $400$.
2. Round $1,872$ to the nearest $100$. The tens digit is $7$. Since $7$ is $5$ or more, $1$ is added to the hundreds digit ($8 + 1 = 9$), and the tens and units digits become $0$. The result is $1,900$.
3. Round $5,050$ to the nearest $100$. The tens digit is $5$. Since $5$ is $5$ or more, $1$ is added to the hundreds digit ($0 + 1 = 1$), and the tens and units digits become $0$. The result is $5,100$.

Rounding to the Nearest 1,000

To round a number to the nearest $1,000$, the hundreds digit must be examined.

The following examples show how to round numbers to the nearest $1,000$:

1. Round $3,245$ to the nearest $1,000$. The hundreds digit is $2$. Since $2$ is less than $5$, the thousands digit remains $3$, and the hundreds, tens and units digits become $0$. The result is $3,000$.
2. Round $8,712$ to the nearest $1,000$. The hundreds digit is $7$. Since $7$ is $5$ or more, $1$ is added to the thousands digit ($8 + 1 = 9$), and the hundreds, tens and units digits become $0$. The result is $9,000$.
3. Round $14,500$ to the nearest $1,000$. The hundreds digit is $5$. Since $5$ is $5$ or more, $1$ is added to the thousands digit ($4 + 1 = 5$), and the hundreds, tens and units digits become $0$. The result is $15,000$.

Approximating Values of Addition and Subtraction

To approximate the result of an addition or subtraction, the numbers are usually rounded to a specified place value (such as the nearest $10$, $100$ or $1,000$) before performing the operation. This provides a quick estimate of the final answer.

Worked Examples: Addition

Example 1: Approximate the sum of $47$ and $82$ by rounding each number to the nearest $10$.

Solution:

$$
\begin{aligned}
47 &\approx 50 \quad (\text{rounded to the nearest } 10) \\
82 &\approx 80 \quad (\text{rounded to the nearest } 10)
\end{aligned}
$$

Now, add the approximated values:

$$50 + 80 = 130$$

The approximate sum is $130$. (The exact sum is $129$).

Example 2: Approximate the sum of $384$ and $519$ by rounding each number to the nearest $100$.

Solution:

$$
\begin{aligned}
384 &\approx 400 \quad (\text{rounded to the nearest } 100) \\
519 &\approx 500 \quad (\text{rounded to the nearest } 100)
\end{aligned}
$$

Now, add the approximated values:

$$400 + 500 = 900$$

The approximate sum is $900$. (The exact sum is $903$).

Example 3: Approximate the sum of $4,230$ and $5,890$ by rounding each number to the nearest $1,000$.

Solution:

$$
\begin{aligned}
4,230 &\approx 4,000 \quad (\text{rounded to the nearest } 1,000) \\
5,890 &\approx 6,000 \quad (\text{rounded to the nearest } 1,000)
\end{aligned}
$$

Now, add the approximated values:

$$4,000 + 6,000 = 10,000$$

The approximate sum is $10,000$. (The exact sum is $10,120$).

Worked Examples: Subtraction

Example 1: Approximate the difference between $93$ and $47$ by rounding each number to the nearest $10$.

Solution:

$$
\begin{aligned}
93 &\approx 90 \quad (\text{rounded to the nearest } 10) \\
47 &\approx 50 \quad (\text{rounded to the nearest } 10)
\end{aligned}
$$

Now, subtract the approximated values:

$$90 – 50 = 40$$

The approximate difference is $40$. (The exact difference is $46$).

Example 2: Approximate the difference between $875$ and $321$ by rounding each number to the nearest $100$.

Solution:

$$
\begin{aligned}
875 &\approx 900 \quad (\text{rounded to the nearest } 100) \\
321 &\approx 300 \quad (\text{rounded to the nearest } 100)
\end{aligned}
$$

Now, subtract the approximated values:

$$900 – 300 = 600$$

The approximate difference is $600$. (The exact difference is $554$).

Example 3: Approximate the difference between $7,840$ and $2,150$ by rounding each number to the nearest $1,000$.

Solution:

$$
\begin{aligned}
7,840 &\approx 8,000 \quad (\text{rounded to the nearest } 1,000) \\
2,150 &\approx 2,000 \quad (\text{rounded to the nearest } 1,000)
\end{aligned}
$$

Now, subtract the approximated values:

$$8,000 – 2,000 = 6,000$$

The approximate difference is $6,000$. (The exact difference is $5,690$).

Approximating the Results of Multiplication and Division

Approximation is highly useful in multiplication and division because exact calculations can be time-consuming. Rounding numbers to the nearest $10$, $100$ or $1,000$ before multiplying or dividing allows for a quick estimation of the product or quotient.

Worked Examples: Multiplication

Example 1: Approximate the product of $23$ and $48$ by rounding each number to the nearest $10$.

Solution:

$$
\begin{aligned}
23 &\approx 20 \quad (\text{rounded to the nearest } 10) \\
48 &\approx 50 \quad (\text{rounded to the nearest } 10)
\end{aligned}
$$

Now, multiply the approximated values:

$$20 \times 50 = 1,000$$

The approximate product is $1,000$. (The exact product is $1,104$).

Example 2: Approximate the product of $192$ and $415$ by rounding each number to the nearest $100$.

Solution:

$$
\begin{aligned}
192 &\approx 200 \quad (\text{rounded to the nearest } 100) \\
415 &\approx 400 \quad (\text{rounded to the nearest } 100)
\end{aligned}
$$

Now, multiply the approximated values:

$$200 \times 400 = 80,000$$

The approximate product is $80,000$. (The exact product is $79,680$).

Example 3: Approximate the product of $8$ and $294$ by rounding the second number to the nearest $100$.

Solution:

$$
\begin{aligned}
8 &\approx 8 \quad (\text{keep single-digit numbers as they are}) \\
294 &\approx 300 \quad (\text{rounded to the nearest } 100)
\end{aligned}
$$

Now, multiply the values:

$$8 \times 300 = 2,400$$

The approximate product is $2,400$. (The exact product is $2,352$).

Worked Examples: Division

Example 1: Approximate the quotient of $84 \div 19$ by rounding each number to the nearest $10$.

Solution:

$$
\begin{aligned}
84 &\approx 80 \quad (\text{rounded to the nearest } 10) \\
19 &\approx 20 \quad (\text{rounded to the nearest } 10)
\end{aligned}
$$

Now, divide the approximated values:

$$80 \div 20 = 4$$

The approximate quotient is $4$. (The exact quotient is $4.42$ to two decimal places).

Example 2: Approximate the quotient of $587 \div 28$ by rounding the dividend to the nearest $100$ and the divisor to the nearest $10$.

Solution:

$$
\begin{aligned}
587 &\approx 600 \quad (\text{rounded to the nearest } 100) \\
28 &\approx 30 \quad (\text{rounded to the nearest } 10)
\end{aligned}
$$

Now, divide the approximated values:

$$600 \div 30 = 20$$

The approximate quotient is $20$. (The exact quotient is $20.96$ to two decimal places).

Example 3: Approximate the quotient of $8,120 \div 395$ by rounding the dividend to the nearest $1,000$ and the divisor to the nearest $100$.

Solution:

$$
\begin{aligned}
8,120 &\approx 8,000 \quad (\text{rounded to the nearest } 1,000) \\
395 &\approx 400 \quad (\text{rounded to the nearest } 100)
\end{aligned}
$$

Now, divide the approximated values:

$$8,000 \div 400 = 20$$

The approximate quotient is $20$. (The exact quotient is $20.56$ to two decimal places).

Applying Approximation in Everyday Life

Approximation is a valuable skill used daily by shoppers, traders, builders and government planners. It helps in making quick estimates when exact calculations are not immediately possible or necessary.

Common situations where approximation is applied include:

1. Market Shopping: A shopper can round the prices of items to the nearest $10$ or $100$ Naira to estimate the total cost and ensure they have enough money before reaching the cashier.
2. Budgeting: Families estimate the cost of school fees, uniforms and books by rounding the costs to the nearest $1,000$ Naira.
3. Travel and Distance: Drivers estimate the distance between towns or the amount of fuel needed by rounding numbers to the nearest $10$ or $100$ kilometres.

Observe the illustration below, which shows a typical market scene where a shopper estimates the total cost of food items by rounding their prices.

Worked Examples: Everyday Life Scenarios

Example 1: A trader in a market buys three bags of beans for $\text{N}28,500$, $\text{N}31,200$ and $\text{N}19,800$. Approximate the total cost to the nearest thousand Naira.

Solution:

First, round each amount to the nearest $1,000$ Naira:

$$
\begin{aligned}
\text{N}28,500 &\approx \text{N}29,000 \\
\text{N}31,200 &\approx \text{N}31,000 \\
\text{N}19,800 &\approx \text{N}20,000
\end{aligned}
$$

Next, add the rounded values together:

$$\text{N}29,000 + \text{N}31,000 + \text{N}20,000 = \text{N}80,000$$

The approximate total cost of the bags of beans is $\text{N}80,000$.

Example 2: A student needs to buy $8$ textbooks for the new school term. Each textbook costs $\text{N}1,950$. Approximate the total cost of the books by rounding the cost of a single book to the nearest thousand Naira.

Solution:

First, round the cost of one book to the nearest $1,000$ Naira:

$$\text{N}1,950 \approx \text{N}2,000$$

Next, multiply the rounded cost by the number of books:

$$\text{N}2,000 \times 8 = \text{N}16,000$$

The approximate total cost of the textbooks is $\text{N}16,000$.

Quantitative Aptitude Involving Approximation

Quantitative aptitude questions test the ability to identify mathematical patterns and apply rules of approximation to solve puzzles. These questions often use unique symbols to represent specific rounding operations.

Worked Examples: Quantitative Aptitude Puzzles

Example 1: Let the symbol $\otimes$ represent an operation where the two numbers are rounded to the nearest $10$ and then added together.

For example: $24 \otimes 37 = 20 + 40 = 60$.

Find the value of $43 \otimes 58$.

Solution:

First, round each number to the nearest $10$:

$$
\begin{aligned}
43 &\approx 40 \\
58 &\approx 60
\end{aligned}
$$

Now, perform the addition:

$$40 + 60 = 100$$

Therefore, $43 \otimes 58 = 100$.

Example 2: Let the bracket $[x]$ represent the value of $x$ rounded to the nearest $100$.

For example: $[249] = 200$ and $[350] = 400$.

Find the value of $[782] – [129]$.

Solution:

First, find the rounded value of each bracketed number to the nearest $100$:

$$
\begin{aligned}
[782] &= 800 \\
[129] &= 100
\end{aligned}
$$

Now, subtract the values:

$$800 – 100 = 700$$

Therefore, $[782] – [129] = 700$.

Example 3: Let the symbol $\Delta$ represent an operation where the first number is rounded to the nearest $1,000$, the second number is rounded to the nearest $100$, and then the first rounded value is divided by the second rounded value.

For example: $6,120 \Delta 195 = 6,000 \div 200 = 30$.

Find the value of $11,850 \Delta 290$.

Solution:

First, round the first number to the nearest $1,000$:

$$11,850 \approx 12,000$$

Next, round the second number to the nearest $100$:

$$290 \approx 300$$

Now, divide the two rounded values:

$$12,000 \div 300 = 40$$

Therefore, $11,850 \Delta 290 = 40$.

Practice Exercises

Section A: Rounding to Specific Place Values

Round the following numbers to the nearest 10:

1. $44$
2. $187$
3. $2,055$

Round the following numbers to the nearest 100:

1. $329$
2. $1,781$
3. $8,050$

Round the following numbers to the nearest 1,000:

1. $4,392$
2. $12,850$
3. $95,100$

Section B: Approximating Basic Operations

Solve the following approximation problems:

1. Approximate the sum of $58$ and $73$ by rounding each number to the nearest $10$.
2. Approximate the difference between $842$ and $319$ by rounding each number to the nearest $100$.
3. Approximate the product of $38$ and $52$ by rounding each number to the nearest $10$.
4. Approximate the quotient of $5,920 \div 295$ by rounding the dividend to the nearest $1,000$ and the divisor to the nearest $100$.

Section C: Everyday Life Applications

Solve the following word problems:

1. A school principal buys three office desks costing $\text{N}12,400$, $\text{N}15,800$ and $\text{N}9,100$. Approximate the total cost of the desks to the nearest thousand Naira.
2. A farmer harvests $48$ baskets of tomatoes. If each basket contains approximately $92$ tomatoes, estimate the total number of tomatoes harvested by rounding both numbers to the nearest $10$.
3. A bus driver travels a distance of $387$ kilometres on Monday and $212$ kilometres on Tuesday. Approximate the total distance travelled over the two days to the nearest hundred kilometres.

Section D: Quantitative Aptitude

Solve the following aptitude puzzles:

1. If $x \oplus y$ means: round $x$ to the nearest $10$, round $y$ to the nearest $10$, and then multiply the results. Find the value of $18 \oplus 42$.
2. If the symbol $\langle a \rangle$ represents the value of $a$ rounded to the nearest $1,000$. Find the value of $\langle 8,750 \rangle – \langle 2,120 \rangle$.
3. If $p \ \square \ q$ means: round $p$ to the nearest $100$, round $q$ to the nearest $10$, and then divide the first rounded value by the second. Find the value of $815 \ \square \ 38$.