Understanding the Problem: Finding the Other Number

The question asks us to find an unknown number when we know the product of this number and another number, and we also know the value of the other number. In mathematical terms, if we have two numbers, let's call them 'Number 1' and 'Number 2', their product is found by multiplying them: Number 1 $\times$ Number 2 = Product.

We are given:

• The product of the two numbers is 0.324.
• One of the numbers is 1.2.
• We need to find the other number.

Setting up the Equation for the Unknown Number

Let the unknown number be represented by the variable $x$. We can write the relationship between the numbers and their product as an equation:

Unknown Number $\times$ Known Number = Product

$x \times 1.2 = 0.324$

To find the value of $x$, we need to isolate it on one side of the equation. We can do this by dividing the product by the known number:

$x = \frac{0.324}{1.2}$

Calculating the Other Number (Division of Decimals)

Now, we need to perform the division $0.324 \div 1.2$. Dividing decimals can sometimes be tricky, so it's often helpful to convert the divisor (the number we are dividing by) into a whole number.

To make 1.2 a whole number, we need to multiply it by 10 (move the decimal point one place to the right). We must do the same to the dividend (the number being divided), which is 0.324, to keep the value of the fraction the same.

• Move the decimal in the divisor (1.2) one place to the right: $1.2 \times 10 = 12$.
• Move the decimal in the dividend (0.324) one place to the right: $0.324 \times 10 = 3.24$.

So, the division becomes:

$x = \frac{3.24}{12}$

Now, we divide 3.24 by 12. We can perform long division:

• Divide 3 by 12. It doesn't go, so write down 0.
• Place the decimal point in the quotient directly above the decimal point in the dividend (3.24).
• Consider 32 divided by 12. $12 \times 2 = 24$. Write down 2 after the decimal point in the quotient. $32 - 24 = 8$.
• Bring down the next digit, 4. We now have 84.
• Divide 84 by 12. $12 \times 7 = 84$. Write down 7 in the quotient. $84 - 84 = 0$.
• The remainder is 0, so the division is complete.

The result of the division $3.24 \div 12$ is 0.27.

Therefore, the other number is 0.27.

Verifying the Answer

To verify our answer, we can multiply the two numbers we have found (0.27 and 1.2) to see if their product is 0.324.

$1.2 \times 0.27$

Let's ignore the decimals for a moment and multiply $12 \times 27$:

$12 \times 27 = 324$

Now, count the total number of decimal places in the original numbers: 1.2 has one decimal place, and 0.27 has two decimal places. In total, there are $1 + 2 = 3$ decimal places.

We need to place the decimal point in 324 so that there are three digits after the decimal point. Starting from the right, we move the decimal point three places to the left: $324 \rightarrow 32.4 \rightarrow 3.24 \rightarrow 0.324$.

The product is 0.324, which matches the product given in the question. Our answer is correct.

Identifying the Correct Option

Based on our calculation, the other number is 0.27. We look at the provided options to find this value.

• Option 1: 2.7
• Option 2: 0.27
• Option 3: 0.027
• Option 4: 27

Option 2 matches our calculated value.

Revision Table: Key Concepts

Additional Information: Related Decimal Operations

Understanding operations with decimals is fundamental in mathematics. Here are a few related concepts:

• Adding/Subtracting Decimals: Align the decimal points vertically and add or subtract as with whole numbers.
• Multiplying Decimals: Multiply the numbers as if they were whole numbers. Then, count the total number of decimal places in the original numbers and place the decimal point in the product so it has that many decimal places.
• Converting Decimals to Fractions: Write the decimal number over a power of 10 corresponding to the number of decimal places (e.g., 0.27 = 27/100, 1.2 = 12/10). This can sometimes help with understanding or calculation.
• Estimating with Decimals: Round the decimal numbers to the nearest whole number or easily manageable decimals to get an approximate answer and check if your calculated answer is reasonable. For example, 0.324 is close to 0.3, and 1.2 is close to 1. 0.3 ÷ 1 = 0.3. Our answer 0.27 is close to 0.3, which is a good sign.