Question

Solve the following. Carry division to the hundredths place where necessary. \(3.75 \div 2.5=\) ______

Short answer

1.5

Step-by-step solution

Convert the Division into a Fraction

To solve the division problem \(3.75 \div 2.5\), first express it as a fraction:\[\frac{3.75}{2.5}\]

Eliminate the Decimal by Multiplying Numerator and Denominator

Both the numerator and the denominator have decimals. To eliminate them, multiply the numerator and the denominator by 10 to make them whole numbers:\[\frac{3.75 \times 10}{2.5 \times 10} = \frac{37.5}{25}\]

Simplify the Fraction

Now, simplify the fraction \(\frac{37.5}{25}\) by dividing both the numerator and the denominator by the greatest common divisor (GCD). Since 37.5 and 25 are decimal numbers after multiplication, let's convert 37.5 to a whole number again by multiplying by 10:\[\frac{375}{250}\] Now, divide both by 25, which is their GCD:\[\frac{375 \div 25}{250 \div 25} = \frac{15}{10}\]

Convert the Fraction to a Decimal

Convert the fraction \(\frac{15}{10}\) to a decimal by performing division:\[15 \div 10 = 1.5\]

Verify and Round the Result

Verify that the division \(1.5\) achieves two decimal places. Since it is already to the hundredths place, no rounding is necessary.

Key concepts

Decimal Conversion

Decimal conversion is crucial when we deal with numbers not perfectly expressed as whole numbers. In this problem, we needed to convert the numbers involved in the division process into a string of digits more comfortable to handle mathematically. When you initially see a division problem like \( 3.75 \div 2.5 \), think of it as breaking \( 3.75 \) into manageable units. This unit breakdown can be more easily processed in fractional form. Converting to decimals often leads to simpler calculations and a clearer answer.

To eliminate the decimals, we multiply each number by 10, transforming \( 3.75 \) into \( 37.5 \) and \( 2.5 \) into \( 25 \). This step doesn't alter the numerical value of the division but adjusts the form for easier computations. It's like peeling away the complex outer layer to reveal the core number ready for operations. Decimal conversion enables us to see numbers from a whole new perspective, allowing for smoother mathematical operations.

Fractions

Fractions represent numbers as part of a whole. In simplifying division problems, expressing numbers as fractions can make calculations more intuitive. For \( 3.75 \div 2.5 \), we start with the fraction \( \frac{3.75}{2.5} \).

But why switch to fractions?

• Fractions allow for a easier transition to multiplication which replaces division as seen in reciprocal methods.
• They cut out potential missteps when dealing with decimals.

By converting decimals to fractions or multiples of base 10, as we did by turning \( 3.75 \) and \( 2.5 \) to \( \frac{37.5}{25} \), the problem morphs into an easier one dealing with integers, eliminating decimal struggle. Keep an eye out for the wonderful flexibility fractions offer in numerical representations, making some complex problems simply melt away.

Simplification

Simplification is about making math easier to process. The act of taking a daunting expression like \( \frac{375}{250} \) and reducing it to its simplest form can vastly ease calculations, diminish errors, and clarify the answer. Simplification means reducing fractions by dividing the numerator and denominator by their greatest common divisor (GCD).

In our problem, \( \frac{375}{250} \) simplified to \( \frac{15}{10} \) after determining and applying the GCD, which was \( 25 \). From there, you can also simplify further to the simplest form \( \frac{3}{2} \) if required, but for decimals, \( \frac{15}{10} \) is adequate to directly find our answer: \( 1.5 \).

Simplifying fractions helps reveal the clearest form of numerical relationships. The process is like decluttering—a necessary step to reach clear understanding and precise results.

Greatest Common Divisor

The greatest common divisor (GCD), or the greatest common factor, is the largest number that divides both the numerator and the denominator without leaving a remainder. Identifying the GCD is crucial in the simplification process.

It helps make a fraction more intelligible by identifying how much each part can be divided equally. In this problem, we determined the GCD of \( 375 \) and \( 250 \) to be \( 25 \). Dividing both the numerator and denominator by the GCD reveals our simplified fraction \( \frac{15}{10} \).

The GCD is like finding the common thread for two numbers, optimizing their forms without altering their meanings. Using the GCD in simplification is an efficient calculative strategy that clarifies the essence of given numbers in a fraction.