Solving the Product of Two Numbers Problem

The problem asks us to find an unknown number when the product of this number and another given number (2.50) is known (40).

Let's break down the problem:

• We are given two numbers.
• Their product (the result when they are multiplied) is 40.
• One of the numbers is 2.50.
• We need to find the other number.

Setting up the Equation

Let the unknown number be represented by the variable \(x\). According to the problem statement, the product of 2.50 and \(x\) is 40.

This can be written as an equation:

\( 2.50 \times x = 40 \)

Solving for the Unknown Number

To find the value of \(x\), we need to isolate \(x\) on one side of the equation. We can do this by dividing both sides of the equation by 2.50.

\( x = \frac{40}{2.50} \)

Performing the Calculation

Now, we perform the division:

\( x = \frac{40}{2.50} \)

To make the division easier, we can remove the decimal from 2.50 by multiplying both the numerator and the denominator by 100:

\( x = \frac{40 \times 100}{2.50 \times 100} \)

\( x = \frac{4000}{250} \)

Now, we can simplify the fraction by dividing both numerator and denominator by 10:

\( x = \frac{400}{25} \)

Finally, perform the division:

\( 400 \div 25 \)

We can think of this as how many quarters (25) are in 400. There are 4 quarters in 100, so there are \(4 \times 4 = 16\) quarters in 400.

Alternatively, using long division:

Divide 40 by 25: 25 goes into 40 once with a remainder of 15. Bring down the 0 to make 150. Divide 150 by 25: 25 goes into 150 six times (\(6 \times 25 = 150\)).

\( x = 16 \)

So, the other number is 16.

Verification

Let's check if the product of 2.50 and 16 is indeed 40:

\( 2.50 \times 16 \)

\( (2 + 0.50) \times 16 = (2 \times 16) + (0.50 \times 16) \)

\( = 32 + 8 \)

\( = 40 \)

The product is 40, which matches the information given in the problem. Therefore, our answer is correct.

Revision Table: Key Concepts for Product Problems

Additional Information: Working with Decimals

When solving equations involving decimals, it can sometimes be helpful to convert the decimals to fractions or multiply by a power of 10 to work with whole numbers. In this problem, we multiplied by 100 to change 2.50 into 250 and 40 into 4000, which made the division \(4000 \div 250\) or \(400 \div 25\) easier.

• Decimal multiplication: Multiply as if they are whole numbers, then place the decimal point in the product based on the total number of decimal places in the original numbers.
• Decimal division: You can make the divisor a whole number by multiplying both the divisor and the dividend by the same power of 10.