Problms involved with real numbers

Problems Involved with
Whole Numbers, Decimals, and
Fractions

How to add whole numbers?
The first step is to line up the numbers vertically so
that the units digits are in the same column. Next,
add the units digits, the tens digits, and the
hundreds digits.

Adding whole numbers
Each number being added is called an addend and the total,
which is the answer to the addition problem is called sum.
Example:
6 addend
+_4__ addend
10 sum

Problem Solving: Adding whole numbers
1. Jose sold 91 apples, 150 oranges and 141 pears. How many pieces
of fruit did he sell altogether?
Solution:
150
+141
_ 91
382
Answer: Jose sold 382 pieces of fruit altogether.

Another problem…
2. Farmer Ben planted 7 992 seeds in the first half of the year and
1 466 in the second half. How many seeds did he plant in the year?
Solution:
7 992
+1 466
9 458
Answer: Farmer Ben planted 9 458 seeds in the year.

How to subtract whole numbers?
Write the smaller number under the larger, taking
care to align the same units. Then, starting with the
ones on the right, subtract each digit on the bottom
from the corresponding digit on top. When the
bottom digit is greater, consider the top digit
increased by 10. To compensate, add 1 to the next
bottom digit.

Subtracting whole numbers
Subtracting whole numbers is the inverse operation of adding whole numbers.
Instead of adding two numbers to get a sum, you are removing one number
from another to get a difference. First, look at the following simple subtraction
problem.
Example:
8 minuend
-4_ subtrahend
4 difference

Problem solving: Subtracting whole numbers
1. Jessica has 1 135 beads. 604 beads are red and the rest are blue.
How many blue beads does she have?
Solution:
1 135
- _604
531
Answer: Jessica have 531 blue beads.

Another problem…
Beth and Ken donated Php 2 300 to a charitable organization. Ken
donated Php 658. How much did Beth donated?
Solution:
2 300
-_ 658_
1 642
Answer: Beth donated Php 1 642 to a charitable organization.

How to multiply whole numbers?
Align the multiplier (on the bottom) with the ones digit
of the multiplicand (on top), and draw a line. Then
multiply each digit of the multiplicand. Write the ones
digit of each product below the line.
If there is a tens digit, carry it -- add it -- to the next
product.

Multiplying whole numbers
The basic idea of multiplication is repeated addition.
Example:
5 factors
x 3_ factors
15 product

Problem Solving: Multiplying whole numbers
1. An apartment has 4 bedrooms. Each bedroom has 3 bookcases.
How many bookcases are there in the apartment?
Solution:
4
_x3_
12
Answer: There are 12 bookcases in the apartment

Another problem…
2. There are 60 minutes in 1 hour. How many minutes are there in
12 hours?
Solution:
60
_x12_
720
Answer: There are 720 minutes in 12 hours.

How to divide whole numbers?
The problem of division is to find what number times
the Divisor will equal the Dividend. That number is
called the Quotient. To find the quotient, there is a
method called short division.

Dividing whole numbers
Division is the inverse of multiplication; therefore it depends on knowing the
multiplication table.
Example:
2 quotient
8 16 dividend
-16__
0
divisor

Problem Solving: Dividing whole numbers
1. In one night, a movie theater sells tickets for 6 450 dollars. Each
ticket costs 15 dollars. How many people purchased a ticket?
Solution: 430
15 6 450
-60___
45
-45__
0
Answer: There are 430 people who purchased a ticket.

Another problem…
2. How many hours are there in 660 minutes?
Solution: 11
60 660
-60__
60
_-60_
0
Answer: There are 11 hours in 660 minutes.

How to add decimals?
To add decimals, follow these steps:
1. Write down the numbers, one under the other, with
the decimal points lined up.
2. Put in zeros so the numbers have the same length
3. Then add using column addition, remembering to
put the decimal point in the answer.

Adding Decimals
Decimal. Based on 10. Example: the numbers we use in everyday life are
decimal numbers, because there are 10 of them (0,1,2,3,4,5,6,7,8 and 9).
Often "decimal number" is also used to mean a number that uses a decimal
point followed by digits that show a value smaller than one.
Example:
8.3 addend
+4.7_ addend
13.0 sum

Problem Solving: Adding Decimals
Problem #1: Ellen wanted to buy the following items: A DVD player for $49.95,
a DVD holder for $19.95 and a personal stereo for $21.95. Does Ellen have
enough money to buy all three items if she has $90 with her?
Analysis: The phrase enough money tells us that we need to estimate the sum
of the three items. We will estimate the sum by rounding each decimal to the
nearest one. We must then compare our estimated sum with $90 to see if she
has enough money to buy these items.
Answer: No, because rounding each decimal to the nearest one, we get an
estimate of $92, and Ellen only has $90 with her.

Another problem…
Problem #2: Melissa purchased $39.46 in groceries at a store. The cashier
gave her $1.46 in change from a $50 bill. Melissa gave the cashier an angry
look. What did the cashier do wrong?
Analysis: We need to estimate the difference to see if the cashier made a
mistake.
$50.00 - $40.00 = $10.00
Estimate: $1.46 is much smaller than the estimated difference of $10.00. So
the cashier must have given Melissa the wrong change.

How to subtract decimals?
To subtract decimals, follow these steps:
1. Write down the two numbers, one under the other,
with the decimal points lined up.
2. Add zeros so the numbers have the same length.
3. Then subtract normally, remembering to put the
decimal point in the answer.

Subtracting Decimals
Subtracting decimals is easy when you keep your work neat.
Example:
67.9 minuend
_-23.2_ subtrahend
44.7 difference

Problem Solving: Subtracting decimals
Problem #1: Drake has 2.5 million in his bank account, he withdraw 1.3 million
to buy a house and lot for his family. How much money does he have left?
Solution:
2.5
-1.3_
1.2
Explanation: The total amount of money Drake has in his bank account is 2.5
million, since he withdrawn 1.3 million to buy a house and lot, the remaining
amount in the bank is now 1.2 million.

How to multiply decimals?
Just follow these steps:
1.Multiply normally, ignoring the decimal points.
2.Then put the decimal point in the answer - it will have
as many decimal places as the two original numbers
combined.

Multiplying Decimals
When you multiply decimals, you multiply them the exact same way you would multiply
whole numbers. Then you count the number of spaces you have in your 2(two) numbers
to multiply and you got to have that many spaces in your product.
Example:
32.12 factor
_x0.5_ factor
16.06 product

Problem solving: Multiplying decimals
Problem #1:Two students multiplied 0.2 by 0.4. Student 1 found a product of
0.8 and Student 2 found a product of 0.08. Which student had the correct
answer? Explain.
Student 1: 0.8 Student 2: 0.08
Analysis: Let's convert each decimal to a fraction to help us solve this problem.
Fractions: 2/10 x 4/10= 8/100
Decimals: 0.2 x 0.4= 0.08
If we multiply two tenths by four tenths, we get a product of eight hundredths.
Answer: Student 2 is correct since 0.2 x 0.4 = 0.08.

How to divide decimals?
To divide decimal numbers:
1. If the divisor is not a whole number, move decimal
point to right to make it a whole number and move
decimal point in dividend the same number of places.
2. Divide as usual. ...
3. Put decimal point directly above decimal point in the
dividend.
4. Check your answer.

Dividing Decimals
The picture above shows how to divide decimals.

Problem Solving: Dividing decimals
School lunches cost $14.50 per week. About how much would 15.5 weeks
of lunches cost?
Analysis: We need to estimate the product of $14.50 and 15.5. To do this,
we will round one factor up and one factor down.
Estimate:
$14.50 $10
x15.5 _20_
Answer: The cost of 15.5 weeks of school lunches would be about $200.

How to add fractions(similar denominators)?
Instructions for adding fractions with the same
denominator
1. Build each fraction (if needed) so that both
denominators are equal.
2. Add the numerators of the fractions.
3. The new denominator will be the denominator of
the built-up fractions.
4. Reduce or simplify your answer, if needed. Factor
the numerator.

How to Add Fractions with Different Denominators?
When the fractions that you want to add have
different denominators, there are a few different ways
you can do it. Here, you’ll learn the easy way, then a
quick trick that works in a few special cases, and
finally, the traditional way.

Here’s some ways to do it:
1. Cross-multiply the two fractions and add the results
together to get the numerator of the answer.
Suppose you want to add the fractions 1/3 and 2/5. To
get the numerator of the answer, cross-multiply. In other
words, multiply the numerator of each fraction by the
denominator of the other.

2. Multiply the two denominators together to get
the denominator of the answer.
To get the denominator, just multiply the
denominators of the two fractions.

3. Write your answer as a fraction.
1 + 2 = _1(5) +_2(3)_ = 5 + 6 = 11
3 5 15 15 15 15 15
When you add fractions, you sometimes need to reduce
the answer that you get. Here’s an example:
Because the numerator and the denominator are both
even numbers, you know that the fraction can be
reduced. So try dividing both numbers by 2:

This fraction can’t be reduced further, so 37/40 is
the final answer.
In some cases, you may have to add more than one
fraction. The method is similar, with one small
tweak.

1. Start out by multiplying the numerator of the first
fraction by the denominators of all the other
fractions.
(1 5 7) = 35

2. Do the same with the second fraction and add
this value to the first.
35 + (3 2 7) = 35 + 42
3. Do the same with the remaining fraction(s).
35 + 42 + (4 2 5) = 35 + 42 + 40 = 117
When you’re done, you have the numerator of the
answer.

4. To get the denominator, just multiply all the
denominators together:
Complete Solution: 1 +3+4 = 1(35) + 3(14) + 4(10)
2 5 7 70 70 70
= 35 + 42 + 40
70 70 70
= 117
70

Problem Solving: Adding Fractions
Example #1: John walked 1/2 of a mile yesterday and 3/4 of a mile today. How
many miles has John walked?
Solution:
This word problem requires addition of fractions.
Choosing a common denominator of 4, we get.
1/2 + 3/4 = 2/4 + 3/4 = 5/4
So, John walked a total of 5/4 miles.

Example #2:
Mary is preparing a final exam. She study 3/2 hours on Friday, 6/4 hours on
Saturday, and 2/3 hours on Sunday. How many hours she studied over the
weekend?
Solution
Choosing a common denominator of 12, we get:
3/2 + 5/4 + 2/3 = 18/12 + 15/12 + 8/12 = 41/12 = 3.42 hours
So, Mary studied a total of 3.42 hours.

How to subtract decimals(similar
denominators)?
There are 3 simple steps to subtract fractions
1. Make sure the bottom numbers (the denominators)
are the same.
2. Subtract the top numbers (the numerators). Put the
answer over the same denominator.
3. Simplify the fraction (if needed).

Subtracting Fractions with dissimilar
denominators.
First of all, when subtracting fractions with different
denominators, the first step in the Rule says that we must
change these fractions so that they have the “same
denominator”.
Here are the steps for subtracting fractions with different
denominators.

So, here are the steps.
1. Build each fraction so that both denominators are
equal. Remember, when subtracting fractions, the
denominators must be equal. So we must complete this
step first. What this really means is that you must find
what is called a Common Denominator. Most of the
time you will be required to work the problem using
what’s called the Least Common Denominator (LCD).
In either case you will build each fraction into an
equivalent fraction.

2. Re-write each equivalent fraction using this new
denominator
3. Now you can subtract the numerators, and keep the
denominator of the equivalent fractions.
4. Re-write your answer as a simplified or reduced
fraction, if needed.

Problem Solving: Subtracting Fractions
Example #1: A recipe needs 3/4 teaspoon black pepper and 1/4 red pepper.
How much more black pepper does the recipe need?
This fraction word problem requires subtraction
Solution:
The fact that the problem is asking how much more black pepper the recipe
needs is an indication that 3/4 is bigger than 1/4
However, it does not hurt to check.
3/4 - 1/4 = 2/4 = 1/2
The black pepper is 1/2 of a teaspoon more than the red pepper.

Example #2:
A football player advances 2/3 of a yard. A second player in the same team
advances 5/4 of a yard. How much more yard did the second player
advance?
Again, we need to perform subtraction to solve this problem.
Solution:
5/4 - 2/3 = 15/12 - 8/12 = 7/12
6/12 is = 1/2, so 7/12 is just a bit more than half.
So, the second player advanced by about half of a yard more.

Thank you for participating
and God bless you all!

Problms involved with real numbers

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Problms involved with real numbers