Arithmetic with Whole Numbersدوره: راهنمای مطالعه و تمرین- تست GRE / فصل: GRE Quantitative Reasoning- Numbers and Operations / درس 2
Arithmetic with Whole Numbers
There are four basic mathematical operations- addition, subtraction, multiplication, and division. These four operations are used in a wide range of everyday skills and are the fundamental blocks of arithmetic.
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There are four operations that can be performed when working with whole numbers. These four operations are addition, subtraction, multiplication, and division . We use the operations in everyday life for a multitude of tasks.
The most basic of these four operations is addition . Addition is the operation that involves calculating the total amount of a represented group. To add a set of numbers together, we first need to line them up vertically by place value. Then, add each column of numbers from right to left. If a column has a sum greater than 9, we will carry the tens place to the next column.
Let’s take a visit to the local mall to visit my good friends Val and Keri. They own a local cookie shop where they are busy getting ready for their customers.
Val and Keri have several orders placed for today and need to know how many cookies they should bake. They have orders of 116 chocolate chip cookies, 524 sugar cookies, and 75 snicker doodle cookies. Val asks Keri to calculate how many cookies they must bake today.
Keri knows that to start adding these numbers together, she must line them up vertically by place value. So, Keri writes down 524, 116, and 75 lined up on the order form.
Keri begins by adding the column on the right. 4 + 6 + 5 equals 15. She knows that since her value is greater than 9, she must carry the tens place to the next column. So, she writes down the 5 and carries the 1 to the next column.
To add the next column, Keri will need to add 7 + 1 + 2 + the 1 that we carried. The sum of these numbers is 11. Again, Keri must carry the tens value since the total is greater than 9.
Next, she needs to add the last column 5 + 1 + the carried 1 equals 7. Keri now knows that they need to bake 715 cookies to fill all of their customer orders.
The next operation is subtraction . Subtraction is the operation that involves taking away value from the total amount. To subtract whole numbers, we will need to line the numbers up by place value from right to left. Next, we will subtract each place value. Occasionally, the value that we are subtracting is not large enough to take away the value. When this happens, we must borrow value from the next place value. When you borrow from the next place value, you are borrowing a set of ten. This set of ten is added to the existing value, and subtraction is continued.
Let’s check back in on Val and Keri to see how they are doing on the cookie order. Val and Keri have cooked all of the chocolate chip and snicker doodle cookies but still need to cook the sugar cookies. As they are preparing to start the sugar cookies, the phone rings. Val answers, and it’s their customer. She tells Val that she needs to reduce her order for sugar cookies. She had originally ordered 524 sugar cookies but needs to subtract 134 of them.
Val now needs to subtract the customer’s original order of 524 sugar cookies by 134. To do this, he must first line up the number from right to left by place value.
To begin, Val starts with the values on the right. 4 minus 4 is 0.
Next, Val needs to subtract 2 minus 3. However, he knows that you cannot take 3 from 2, so he must borrow. So, he borrows a set of ten from the next place value. As he borrows a set from the 5, its value becomes a 4. The set of ten also increases the value of the 2 to 12.
Val can now subtract, 12 minus 3 is 9, and 4 minus 1 is 3. Val now knows that he only needs to cook 390 sugar cookies to fill the customer’s order.
Multiplication is another one of the four operations. Multiplication is the process of adding multiple sets of the same number. To multiply, we will take each place value and multiply it to every value in the other number. If the number being multiplied has more than one value, you must add a zero as a place holder each time you move to the next place value. This process continues until all of the places have been multiplied.
Back in the cookie shop, Val and Keri have finished their daily order. As they talk, Keri wonders if they cooked this many cookies daily, how many cookies would they cook in 126 days. To do this, she will need to multiply the daily cookie order of 715 by 126 days.
To start this problem, Keri needs to multiply the 6 by each of the values on top from right to left. If a value is greater than 9, the tens place will be carried to the next place and added to the multiplied value.
Six times 5 is 30, so we keep the zero and carry the 3.
Next, we multiply 6 times 1, which is 6, and then add the carried three to equal 9. The value 9 is written below.
Then, we multiply the 6 times the 7, which is 42. Since there are no additional places on top to be multiplied, we write the 42 below.
Next, we must multiply each place on top by the 2. Since we moved one place on the bottom, we will need to add a zero to the next line below. As we multiply the 2 to each value on top, we get 14300.
As we move to the next place to multiply, we must now add two zeros to the bottom because we have moved over two place values.
Next, we need to multiply the 1 by each value on the top, which gives us 71500.
The last step to multiplying is to add these values together from right to left. After adding these values, the product is 90,090. Keri knows that if they sell 715 cookies for 126 days, they will sell 90,090 cookies.
The last of the four operations is division . Division is the process of taking a total value and dividing it into equal parts. The first step is to set up the problem. The value that is being divided is called the dividend, and the value that we are dividing into is called the divisor.
To set up a division problem, you will use the format dividend divided by divisor.
When dividing, we take the divisor and divide it into each value of the dividend from left to right. The number of times that the value will divide into the dividend goes on top. The value is then multiplied and subtracted from the dividend. The process continues until all of the places in the dividend have been divided. If there is a value that remains at the end, this is called your remainder.
The wonderful smell of cookies has now attracted customers to the cookie shop. Val and Keri are glad to see such a huge crowd. There are 12 customers that have gathered at the cookie counter. Val and Keri know that they have cooked only 715 cookies. Val wants to know how many cookies each of the 12 customers order.
In this problem, 715 is the dividend, and 12 is the divisor. The problem would be set up as 715 divided by 12.
The first step is to take the 12 and see which value it will divide into. 12 will not divide into 7, so we must use the first two values 71. Twelve will go into 71 five times. So, we write the 5 on top.
Next, we need to multiply the 5 times 12 which is 60. This value gets written below the 71 in the dividend.
Now, we must subtract 71 minus 60, which is 11. We must also bring down the next value from the dividend.
The new dividend is 115. Next, we must see how many times 12 will divide into 115, which is 9. This 9 gets written on top.
At this point, we repeat the same process as in the previous step. We need to multiply 9 times the 12, which is 108, which will be written below the 115.
Subtract 108 from 115, which is 7. Since we are out of values in the dividend, 7 is our remainder. Val can now see that he can sell each customer 59 cookies, and he will have 7 cookies leftover.
There are four operations that can be performed when working with whole numbers. These four operations are addition, subtraction, multiplication, and division .
The most basic of these four operations is addition . Addition is the operation that involves calculating the total amount of a represented group.
The next operation is subtraction . Subtraction is the operation that involves taking away value from the total amount.
Multiplication is another one of the four operations. Multiplication is the process of adding multiple sets of the same number.
The last of the four operations is division . Division is the process of taking a total value and dividing it into equal parts.
You will be able to define and use addition, subtraction, multiplication and division in math problems after completing this lesson.
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