Dividing a small quantity by a giant quantity can appear to be a frightening job, however with the precise method, it may be made a lot less complicated. This text will present a step-by-step information on the best way to divide a small quantity by a giant quantity, breaking down the method into manageable chunks. Whether or not you are a scholar scuffling with lengthy division or an grownup seeking to brush up in your math expertise, this text will give you the instruments you have to confidently deal with this mathematical operation.
Step one in dividing a small quantity by a giant quantity is to arrange the issue appropriately. Write the small quantity because the numerator and the massive quantity because the denominator. For instance, if you wish to divide 12 by 24, you’ll write it as 12 ÷ 24. After getting arrange the issue appropriately, you’ll be able to start the division course of. Begin by dividing the primary digit of the numerator by the primary digit of the denominator. In our instance, this might be 1 ÷ 2, which equals 0. Write the 0 above the numerator.
Subsequent, multiply the denominator by the quotient you simply discovered and subtract the outcome from the numerator. In our instance, this might be 2 × 0, which equals 0. We then subtract 0 from 12, which supplies us 12. Carry down the subsequent digit of the numerator and repeat the method. In our instance, this might be 12 ÷ 2, which equals 6. Write the 6 above the numerator. Proceed this course of till there are not any extra digits left within the numerator. In our instance, this might be 12 ÷ 2, which equals 6. We’d then write the 6 above the numerator and the rest could be 0.
Divide Utilizing Lengthy Division
Lengthy division is a technique for dividing massive numbers by smaller numbers. It includes repeated subtraction and multiplication to regularly cut back the dividend (the quantity being divided) till there isn’t any the rest or the rest is smaller than the divisor (the quantity dividing into the dividend).
Listed here are the steps concerned in lengthy division:
Step 1: Set Up the Downside
Write the dividend and the divisor as a fraction, with the dividend because the numerator and the divisor because the denominator. If crucial, multiply or divide each numbers by an element of 10, 100, or 1000 to make the divisor a complete quantity.
Step 2: Discover the First Digit of the Quotient
Divide the primary digit of the dividend by the primary digit of the divisor to seek out the primary digit of the quotient. Write the quotient above the dividend, instantly above the digit being divided.
Step 3: Multiply and Subtract
Multiply the divisor by the quotient digit you simply discovered. Subtract the outcome from the primary a part of the dividend. Carry down the subsequent digit of the dividend.
Step 4: Repeat Steps 2-3
Proceed dividing, multiplying, and subtracting till there are not any extra digits within the dividend. If there’s a the rest, it ought to be smaller than the divisor.
Step 5: Examine Your Reply
To examine your reply, multiply the quotient by the divisor and add the rest. The outcome ought to be the identical as the unique dividend.
Estimate the Quotient
When dividing a small quantity by a giant quantity, the quotient (the reply) shall be a small quantity. To estimate the quotient, divide the primary digit of the dividend (the quantity you are dividing) by the primary digit of the divisor (the quantity you are dividing by). This provides you with an estimate of the quotient.
For instance, for example we need to divide 12 by 100. The primary digit of 12 is 1 and the primary digit of 100 is 1. Dividing 1 by 1 offers us 1, so we estimate that the quotient shall be round 1.
This estimate can be utilized to examine your reply if you really carry out the division. In case your reply is considerably completely different from the estimate, you’ll have made a mistake in your division.
Instance
Let’s divide 12 by 100 utilizing lengthy division:
| 12 |
|---|
| 100 |
| __ |
| 120 |
| -100 |
| 20 |
| -20 |
| 0 |
As you’ll be able to see, the quotient is 0.12, which is near our estimate of 1.
Use Partial Quotients
Partial quotients is a technique for lengthy division that can be utilized to divide a small quantity by a giant quantity. It’s a systematic course of that may be damaged down right into a collection of steps.
Step 1: Arrange the issue
Step one is to arrange the issue. This includes writing the dividend (the quantity being divided) and the divisor (the quantity dividing) in an extended division format. For instance, if we’re dividing 12345 by 678, we’d write it as follows:
| 12345 | | 678 |
Step 2: Discover the primary partial quotient
The following step is to seek out the primary partial quotient. That is the most important digit that may be divided evenly into the primary digit of the dividend. In our instance, the primary digit of the dividend is 1, and the most important digit that may be divided evenly into 1 is 0. We due to this fact write 0 above the lengthy division drawback, as follows:
| 12345 | | 678 |
| 0 |
Step 3: Multiply the divisor by the partial quotient and subtract the outcome from the dividend
The following step is to multiply the divisor by the partial quotient and subtract the outcome from the dividend. In our instance, we’d multiply 678 by 0 and subtract the outcome (which is 0) from the dividend. This leaves us with the next:
| 12345 | | 678 |
| 0 | |
| 12345 |
Step 4: Repeat steps 2 and three till the dividend is zero
We then repeat steps 2 and three till the dividend is zero. In our instance, we’d discover the subsequent partial quotient, multiply the divisor by the partial quotient, and subtract the outcome from the dividend. We’d then proceed this course of till the dividend is zero. The ultimate outcome could be as follows:
| 12345 | | 678 |
| 18 | |
| 0 |
Convert to Fractions
Changing a small quantity to a fraction with a big denominator is a helpful method for making it simpler to divide. To do that, merely add a decimal level to the small quantity after which add as many zeros as wanted to create a denominator of the specified measurement. For instance, to transform 5 to a fraction with a denominator of 100, we’d write 5.00. Dividing 5.00 by 100 would then be equal to dividing 5 by 100, which is far simpler to calculate.
Here’s a desk exhibiting the best way to convert small numbers to fractions with completely different denominators:
| Small Quantity | Fraction |
|---|---|
| 5 | 5.00/100 |
| 10 | 10.00/100 |
| 15 | 15.00/100 |
| 20 | 20.00/100 |
| 25 | 25.00/100 |
After getting transformed the small quantity to a fraction, you’ll be able to then divide it by the big quantity utilizing the usual division algorithm. For instance, to divide 5 by 100, you’ll:
- Arrange the division drawback as follows:
- Divide the primary digit of the dividend (5) by the divisor (100) and write the outcome (0) above the dividend.
- Multiply the divisor by the quotient (0) and write the outcome (0) under the dividend.
- Subtract the outcome from the dividend to get a the rest of 5.00.
- Carry down the subsequent digit of the dividend (0) and repeat steps 2-4 till there are not any extra digits to convey down.
- The ultimate quotient is 0.05, which is equal to five/100 or 0.05 in decimal type.
100 | 5.00
100 | 5.00
0
100 | 5.00
0
0
100 | 5.00
0
0
5.00
100 | 5.00
0
0
5.00
500
Use a Calculator
In case you have a calculator, dividing a small quantity by a giant quantity is simple. Merely enter the dividend (the smaller quantity) and the divisor (the larger quantity) into the calculator, after which press the division key. The calculator will show the quotient (the results of the division).
For instance, if you wish to divide 12 by 3, you’ll enter 12 into the calculator, then press the division key, then enter 3, after which press the equals key. The calculator would show the reply, which is 4.
You may also use a calculator to divide a decimal quantity by a complete quantity. For instance, if you wish to divide 1.2 by 3, you’ll enter 1.2 into the calculator, then press the division key, then enter 3, after which press the equals key. The calculator would show the reply, which is 0.4.
If you wish to divide a complete quantity by a decimal quantity, you’ll be able to convert the decimal quantity to a fraction after which divide. For instance, if you wish to divide 12 by 0.5, you’ll be able to convert 0.5 to the fraction 1/2. Then, you’ll be able to divide 12 by 1/2 by multiplying 12 by the reciprocal of 1/2, which is 2. The reply is 24.
| Dividend | Divisor | Quotient |
|---|---|---|
| 12 | 3 | 4 |
| 1.2 | 3 | 0.4 |
| 12 | 0.5 | 24 |
Resolve Phrase Issues
Division phrase issues usually contain real-world situations the place you have to divide a amount into equal elements or discover the variety of occasions one amount is contained inside one other. To unravel these issues, observe these steps:
- Learn the issue fastidiously to establish the data given.
- Decide what you have to discover, normally represented by the unknown amount (e.g., “What number of luggage?” or “What’s the size?”).
- Arrange a division equation utilizing the given info and the unknown amount.
- Resolve the equation by dividing the dividend by the divisor to seek out the unknown amount.
- Examine your reply by substituting it again into the unique drawback and verifying if it is smart.
Instance 1: Dividing Sweet Evenly
Given 24 items of sweet, what number of luggage are you able to fill if every bag can maintain 3 candies?
- Unknown: Variety of luggage
- Division equation: Variety of luggage = 24 candies / 3 candies per bag
- Fixing: 24 / 3 = 8
- Reply: 8 luggage
Instance 2: Discovering the Size of Fence
In case you have 120 ft of fence and need to enclose a sq. space, what’s the size of every facet of the sq.?
- Unknown: Facet size of sq.
- Division equation: Perimeter = 4 x Facet size, so Facet size = Perimeter / 4
- Fixing: 120 ft / 4 = 30 ft
- Reply: 30 ft per facet
Instance 3: Calculating Distance Traveled
A automobile travels 360 miles in 6 hours. What was the automobile’s common velocity in miles per hour?
- Unknown: Common velocity
- Division equation: Common velocity = Distance / Time
- Fixing: 360 miles / 6 hours = 60 miles per hour
- Reply: 60 miles per hour
Examine Your Reply
After getting discovered a quantity that offers you your denominator, multiply that quantity by your numerator to double examine your reply. If the reply matches your dividend, then you might have efficiently divided the small quantity by the massive quantity. If not, then you will want to attempt once more.
8. Divide 12 by 19,291
To unravel this drawback, arrange your lengthy division such as you would when dividing 12 by 192. Then, to seek out the primary digit of your reply, you multiply 192 by X. As x goes up, so will the results of 192 x. While you get to 192 multiplied by 10, you realize that 19200 is just too excessive (19200 > 12), whereas 192 multiplied by 9 is just too low (192 x 9 = 17280 < 12). Due to this fact, the reply is 192 x 9 = 17280. Subtract 17,280 from 12,000 to get 4800. Carry down the subsequent digit 0, then repeat the method till there are not any extra digits in your dividend.
Setting this all up in lengthy division format ought to provide the following:
0.0006278 19,291)12.0000 115746 48240 38582 96580 96455 1250 Frequent Errors to Keep away from
1. Avoiding Repeated Subtraction
When dividing a small quantity by a big quantity, it is tempting to make use of repeated subtraction. This technique is very inefficient and susceptible to errors. It is higher to make use of the lengthy division technique as a substitute.
2. Misplacing the Decimal Level
Pay shut consideration to the position of the decimal level when dividing a decimal by a complete quantity or one other decimal. Misplacing the decimal can result in incorrect outcomes.
3. Utilizing a Division Signal as a Fraction Bar
The division signal (÷) shouldn’t be the identical as a fraction bar. When dividing a quantity, write it as a numerator and denominator in fraction type or use the lengthy division technique.
4. Forgetting to Embody a The rest
When dividing a small quantity by a big quantity, there could also be a the rest that’s lower than the divisor. This the rest ought to be included within the quotient as a decimal or fraction.
5. Rounding Off Too Early
When dividing a small quantity by a big quantity, it is vital to hold out sufficient decimal locations to realize the specified accuracy. Rounding off too early can result in lack of precision.
6. Dividing Zero by a Quantity
Dividing zero by any quantity (besides zero) leads to undefined. It is because any quantity multiplied by zero is zero, so there isn’t any quantity that may be multiplied by zero to get a non-zero outcome.
7. Dividing a Optimistic Quantity by a Adverse Quantity
Dividing a optimistic quantity by a unfavorable quantity leads to a unfavorable quotient. Equally, dividing a unfavorable quantity by a optimistic quantity leads to a optimistic quotient.
8. Signal Errors in Remainders
When the dividend and divisor have completely different indicators, the signal of the rest would be the similar because the signal of the dividend.
9. Misinterpreting Incomplete Quotients
Incomplete quotients can happen when the divisor is considerably bigger than the dividend. In such instances, the quotient ought to be interpreted as an approximation of the true quotient. To acquire a extra correct quotient, it is necessary to hold out extra decimal locations or use different strategies comparable to a calculator or laptop software program.
Mistake Description Instance Avoiding Repeated Subtraction Utilizing repeated subtraction as a substitute of lengthy division Dividing 1 by 100 utilizing repeated subtraction: 1 – 0.01 – 0.001 – 0.0001 – … Misplacing the Decimal Level Incorrectly putting the decimal level when dividing decimals Dividing 0.5 by 5 and putting the decimal level after the primary digit: 0.10 Utilizing a Division Signal as a Fraction Bar Treating the division signal as a fraction bar Writing 1 ÷ 2 as 1/2, which is a fraction Forgetting to Embody a The rest Omitting the rest when dividing with a decimal divisor Dividing 1 by 3 and ignoring the rest of 1: 0.3 Rounding Off Too Early Untimely rounding of the quotient Dividing 1 by 7 and rounding to 2 decimal locations: 0.14, as a substitute of 0.1428 Dividing Zero by a Quantity Trying to divide zero by a non-zero quantity Dividing 0 by 5: undefined Dividing a Optimistic Quantity by a Adverse Quantity Incorrect signal of the quotient when dividing a optimistic quantity by a unfavorable quantity Dividing 5 by -2: -10, as a substitute of 5 Signal Errors in Remainders Incorrect signal of the rest when the dividend and divisor have completely different indicators Dividing -5 by 2: -2 the rest 1, as a substitute of -2 the rest -1 Misinterpreting Incomplete Quotients Mistaking an incomplete quotient for the true quotient Dividing 1 by 1000: 0.001, as a substitute of an approximation like 0.00099 Follow Makes Excellent
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10. Division Algorithm and Lengthy Division Course of
The division algorithm offers a scientific method to divide small numbers by massive numbers. It includes the next steps:
- Divide the dividend (the small quantity) by the divisor (the big quantity) till the quotient (the outcome) is smaller than the divisor.
- Multiply the divisor by the quotient to get the product.
- Subtract the product from the dividend to get the rest.
- If the rest is zero, the division is full. In any other case, repeat steps 1-3 till the rest is zero or the quotient reaches the specified stage of precision.
The lengthy division course of is an in depth illustration of the division algorithm. It includes establishing the dividend and divisor vertically, performing the division steps (dividing, multiplying, subtracting, and bringing down the subsequent digit), and persevering with till the specified result’s obtained. A step-by-step instance of lengthy division is offered under:
Instance: Clarification: 1256 ÷ 7 Dividend (1256) and divisor (7) 179 R 3 Quotient (179), the rest (3) How To Divide A Small Quantity By A Huge Quantity
When dividing a small quantity by a giant quantity, it is vital to keep in mind that the quotient (the reply) shall be a small quantity as effectively. To carry out this division, you should utilize the next steps:
- Arrange the division drawback with the dividend (the small quantity) on prime and the divisor (the massive quantity) on the underside.
- Divide the primary digit of the dividend by the divisor. If the result’s a decimal, truncate it to the closest complete quantity.
- Multiply the outcome by the divisor and subtract it from the dividend. Carry down the subsequent digit of the dividend.
- Repeat steps 2 and three till you might have introduced down all of the digits of the dividend.
- The quotient is the quantity you might have been writing above the dividend.
For instance, to divide 12 by 100, you’ll arrange the issue as follows:
“`
12 ÷ 100
“`Then, you’ll divide the primary digit of the dividend (1) by the divisor (100). The result’s 0.01, which you’d truncate to 0.
“`
12 ÷ 100 = 0
“`Subsequent, you’ll multiply the outcome (0) by the divisor (100) and subtract it from the dividend (12). This offers you 12.
“`
12 – (0 x 100) = 12
“`You’ll then convey down the subsequent digit of the dividend (2) and repeat steps 2 and three.
“`
122 ÷ 100 = 0.01
“`
“`
122 – (0 x 100) = 122
“`
“`
1222 ÷ 100 = 0.01
“`
“`
1222 – (0 x 100) = 1222
“`The quotient is 0.012, which you’ll be able to write as 0.012 or 1.2%.
Folks additionally ask
How do you divide a fraction by a complete quantity?
To divide a fraction by a complete quantity, you’ll be able to multiply the fraction by the reciprocal of the entire quantity. The reciprocal of a quantity is 1 divided by the quantity.
How do you divide a combined quantity by a complete quantity?
To divide a combined quantity by a complete quantity, you’ll be able to first convert the combined quantity to an improper fraction. An improper fraction is a fraction the place the numerator is larger than or equal to the denominator.
How do you divide a decimal by a complete quantity?
To divide a decimal by a complete quantity, you’ll be able to transfer the decimal level within the dividend (the quantity being divided) to the precise by the identical variety of locations as there are zeros within the divisor (the quantity dividing into the dividend). Then, divide as common.
