Divide Multi-Digit Decimals

Lesson 2 of 2

In this lesson:

• Divide a decimal by a whole number — point goes up first
• Divide a decimal by a decimal — shift both points equally
• Use estimation to confirm where the point belongs

What You Will Be Able to Do

By the end, you will:

1. Divide a decimal by a whole number using long division
2. Divide a decimal by a decimal by shifting both points
3. Estimate to confirm the answer's magnitude

A Pharmacist Splits a Decimal Volume

A pharmacist measures mL of solution into equal doses.

How many milliliters per dose?

You already know long division for whole numbers. The dividend has a decimal point now — does the algorithm still work?

Recall: Long Division for Whole Numbers

The four moves you already know:

1. Divide — how many times does 5 go into 35?
2. Multiply —
3. Subtract —
4. Bring down — (no more digits; done)

Decimal Point Goes UP FIRST

Place the point in the quotient first, then divide as usual.

Worked:

Set up. Place the point. Then divide:

• into is , remainder
• Bring down : into is , remainder
• Bring down : into is , remainder

Quotient: . Estimate confirms.

Leading Zero in the Quotient

• doesn't fit into — write
• doesn't fit into — write
• into is
• into is

The leading in is placed, not skipped.

Append Zeros to Keep Dividing

When the dividend's decimal portion runs out, append zeros — they don't change the value.

• The appended zeros let the algorithm continue
• Stop when the remainder is or when enough places have been computed

Worked:

Extend dividend to :

• into : , r
• Bring down → : , r
• Bring down → : , r
• Bring down → : , r

Quotient . Estimate .

Quick Check:

Estimate first, then long-divide.

Pause. Estimate. Compute. Then advance.

What If the Divisor Is a Decimal?

How do you handle ?

The standard long-division algorithm needs a whole-number divisor.

What would make this fit the algorithm we just used?

Algorithm Needs a Whole Divisor

The divisor must be a whole number for long division to work cleanly.

• is not whole
• We need to transform the problem first
• The transformation must not change the quotient

Shift Both Points the Same Number of Places

. Quotient unchanged.

Why Shifting Both Points Works

Multiplying numerator and denominator by the same number leaves the value unchanged.

The "shift" is just bookkeeping for that fact.

Worked:

Shift both points two places: .

Long-divide: .

Estimate confirms answer is in the s or s — not , not .

Worked:

Shift both points one place: .

Long-divide: .

Estimate — close enough.

Worked:

Whole-number dividend? Write as , then shift two places: .

Long-divide: .

Estimate matches.

Find the Error in This Solution

Shifting only the divisor leaves the quotient wrong by .

Mixed: Which Need the Shift?

Two problems — set up each before computing:

• (no shift; just point up first)
• (shift one place: )

Recognize the divisor type before applying the algorithm.

Your Turn:

Recognize, shift, divide, check against estimate.

Pause. Work through it. Then advance.

Answers and the Common Errors to Watch

• (estimate )
• (estimate )

Watch out:

• Shifting only the divisor
• Skipping a leading zero
• Placing quotient point too late

Key Takeaways from Decimal Division

✓ Quotient's decimal point goes UP FIRST
✓ Append zeros to finish the division
✓ Decimal divisor: shift both points equally
✓ Estimate first, every time

Watch out:

• Don't place the quotient's point last
• Don't shift only one number
• Don't skip a leading zero

Coming Up Next: Factors and Multiples

Next: 6.NS.B.4 — greatest common factor and least common multiple. A turn from algorithm to number-theory reasoning.

Beyond that, your decimal fluency carries forward to 6.RP.A (ratios, percent) and 7.NS.A (rational arithmetic) — every percent calculation is a decimal multiplication; every unit rate is a decimal division.