Question

Solve the following. Carry division to the hundredths place where necessary. \(15 \times 2.08=\) ______

Short answer

15 times 2.08 equals 31.20.

Step-by-step solution

Set up the multiplication

To solve the problem \(15 \times 2.08\), you need to treat \(2.08\) as a number with two decimal places. We'll multiply as if there are no decimals at first and adjust later.

Multiply ignoring the decimal

Multiply \(208\) (ignoring the decimal) by \(15\). Break it down: \(208 \times 15 = 208 \times (10 + 5) = (208 \times 10) + (208 \times 5)\). Calculate these products individually.

Calculate partial products

Compute \(208 \times 10 = 2080\) and \(208 \times 5 = 1040\). Add these results together: \(2080 + 1040 = 3120\).

Place the decimal

Since \(2.08\) has two decimal places, you need to place the decimal point two places from the right in the product you obtained: \(3120\). Therefore, the final result is \(31.20\).

Key concepts

Multiplication Steps

When multiplying numbers, especially decimals, it's essential to follow a clear sequence. This ensures accuracy and helps in understanding the process better. To begin with, scope out the entire exercise before diving into each step. Firstly, set up the multiplication so that the numbers are aligned correctly, typically with the larger number on top for ease. In the original exercise, we multiplied 15 by 2.08, treating 2.08 as 208, ignoring the decimal for simplicity.
Next, break the larger number into more digestible parts. For instance, break down 208 into components that are easier to handle, such as its constituent place values, if needed. Here we divided it into steps of multiplying first by 10 and then by 5. This staged approach allows for a focus on one small multiplication problem at a time, which can be especially helpful when dealing with decimals or larger numbers.

• Set Up: Properly align numbers for easy multiplication.
• Simplify: Break down complex numbers into manageable parts.
• Multiply: Carefully execute each multiplication operation.

Each step builds on the previous one, so maintaining focus and precision is key.

Partial Products

Understanding partial products involves breaking down each multiplication into smaller, simpler calculations. This method is extremely useful when multiplying larger numbers or decimals. As seen with the multiplication of 208 by 15, partial products help streamline the process.
Start by decomposing one of the numbers into easier parts. In our case, 208 can be regarded as a two-part operation, first multiplying by 10 and then by 5. Each operation produces a partial product that needs to be computed individually.
For example, calculating the partial product of 208 multiplied by 10 yields 2080. Contrastingly, 208 multiplied by 5 yields 1040. Writing and summing these partial products gives you the combined result of the multiplication.

• Decompose: Break numbers into simpler multipliers.
• Calculate: Multiply the simpler numbers to get partial products.
• Combine: Sum all partial products for the final result.

By mastering partial products, you gain a powerful tool for tackling multiplication problems with confidence.

Decimal Placement

One of the trickiest parts of multiplying with decimals is knowing where to place the decimal in the final result. It's crucial to remember the original positioning of the decimals involved in the multiplication.
In the step-by-step process for multiplying 15 by 2.08, the decimal was initially ignored to simplify the problem. After calculating the product without considering the decimal, it was placed back into the result by counting the total decimal places in the original numbers. Since 2.08 comprises two decimal places, the final product of 3120 must be adjusted to reflect this, resulting in 31.20.
To accurately position the decimal:

• Count Decimals: Determine the total number of decimal places in the numbers being multiplied.
• Ignore initially: Solve the multiplication as if the numbers are integers to streamline the process.
• Adjust: Place the decimal in the product according to the initial count to achieve the correct result.

Being precise with decimal placement ensures that the magnitude of your answer matches the original numbers, which is crucial in maintaining mathematical accuracy.