Download Sample Gateway Problems: Working with Fractions and the Order of

Math TLC (Math 010 and Math 110)
How to Solve Gateway Problems
Feb-09
NOTE: Gateway problems 1 & 2 on
adding and subtracting fractions can both
be done using the same set of steps.
Adding fractions and subtracting fractions
both require finding a least common
denominator (LCD), which is most easily
done by factoring the denominator
(bottom number) of each fraction into a
product of prime numbers (a number that
can be divided only by itself and 1.)
Sample Gateway Problems:
.
.
Working with Fractions
and the Order of Operations
Without Using a Calculator
1
2
Sample Problem #1 (continued)
Step 3: Multiply the numerator (top)and denominator of each fraction
by the factor(s) needed to turn each denominator into the LCD.
Sample Problem #1: Adding fractions
LCD = 2∙5∙7
Step 1: Factor the two denominators into prime factors, then
write each fraction with its denominator in factored form:
10 = 2∙5
and
35 = 5∙7,
so
.
3 + 2 = 3 + 2
10 35 2∙5 5∙7
Step 2: Find the least common denominator (LCD):
LCD = 2∙5∙7
3
.
3∙7 + 2 ∙2
2∙5∙7
5∙7∙2
.
Step 4: Multiply each numerator out, leaving the denominators in
factored form, then add the two numerators and put them over the
common denominator.
21 + 4 = 21 + 4 = 25 (note that 5∙7∙2 = 2∙5∙7 by
2∙5∙7 5∙7∙2
2∙5∙7
2∙5∙7
the commutative property)
Step 5: Now factor the numerator, then cancel any common factors
that appear in both numerator and denominator. Once you multiply
out any remaining factors, the result is your simplified answer.
= 25 = 5∙5 = /5∙5 = 5 = 5
/
2∙5∙7 2∙5∙7 2∙5∙7
2∙7
14
.
4
1
Math TLC (Math 010 and Math 110)
How to Solve Gateway Problems
Feb-09
Sample Problem #2: Subtracting fractions
Full Solution to Sample Problem #1:
5
14
Step 1: Factor the two denominators into prime factors, then
write each fraction with its denominator in factored form:
Here is the work we expect to see on your worksheet:
10 = 2∙5 and 35 = 5∙7,
14 = 2∙7
so 3 + 2 = 3 + 2 , and LCD = 2∙5∙7
10 35 2∙5 5∙7
/
3 + 2 = 3∙7 + 2 ∙2 = 21 + 4 = 25 = 5∙5 = 5∙5
= 5 = 5
/
2∙5 5∙7 2∙5∙7 5∙7∙2 2∙5∙7 2∙5∙7 2∙5∙7 2∙5∙7 2∙5∙7
2∙7 14
and
35 = 5∙7,
so
5 - 2
2∙7
5∙7
Step 2: Find the least common denominator (LCD):
LCD = 2∙7∙5
5
Sample Problem #2 (continued)
Step 3: Multiply the numerator and denominator of each fraction by
the factor(s) needed to turn each denominator into the LCD: form:
LCD = 2∙7∙5
5∙5
2∙7∙5
6
Full Solution to Sample Problem #2:
3
10
- 2 ∙2
5∙7∙2
Step 4: Multiply out the numerators, leaving the denominators in
factored form, then add the two numerators and put them over the
common denominator.
25 - 4
= 25 - 4 = 21
2∙5∙7 5∙7∙2
2∙5∙7 2∙5∙7
Here is the work we expect to see on your worksheet:
.
Step 5: Now factor the numerator, then cancel any common factors
that appear in both numerator and denominator. Once you multiply
out any remaining factors, the result is your simplified answer.
21 = 3∙7 = 3∙7/ = 3 = 3
2∙5∙7 2∙5∙7 2∙5∙7
10
/ 2∙5
.
14 = 2∙7 and 35 = 5∙7,
so 5 - 2 = 5 - 2 , and LCD = 2∙5∙7
14 35 2∙7 5∙7
/ = 3 = 3
5 - 2 = 5∙5 - 2 ∙2 = 25 - 4 = 21 = 3∙7 = 3∙7
2∙7 5∙7 2∙7∙5 5∙7∙2 2∙5∙7 2∙5∙7 2∙5∙7 2∙5∙7 2∙5∙7
/ 2∙5 10
.
7
8
2
Math TLC (Math 010 and Math 110)
How to Solve Gateway Problems
Feb-09
Sample Problem #3: Multiplying fractions
NOTE: Gateway problems 3 & 5 on
multiplying and dividing fractions can both
be done using the similar steps.
Neither multiplying fractions nor dividing
fractions requires finding an LCD. These
kinds of problems can be most easily done
by factoring both the numerator (top
number) and denominator of both
fractions into a product of prime numbers,
and then canceling any common factors
(numbers that appear on both the top and the bottom.)
9
Sample Problem #3 (continued)
First fraction:
Second fraction:
39= 3∙13 and 50 = 2∙5∙5
15= 3∙5 and 26 = 2∙13
So you can write 39 • 15 as 3∙13 • 3∙5
50 26
2∙5∙5 2∙13
NOTE: You do NOT need an LCD when multiplying fractions.
10
Full Solution to Sample Problem #3:
Step 2: Now just cancel any common factors that appear in
both numerator and denominator. Once you multiply out
any remaining factors, the result is your simplified answer.
/ • 3∙5
/ = 3∙3 = 9
3∙13
/ 2∙13
/
2∙5∙5
2∙5∙2 20
Step 1: Factor both the numerators and denominators into
prime factors, then write each fraction in factored form:
9
20
.
.
Here is the work we expect to see on your worksheet:
NOTE: It is much easier to factor first and then cancel, rather
than multiplying out the numerators and denominators and then
trying to simplify the answer (especially if you aren’t using a
calculator!) If you multiplied first, you’d have gotten
585 , which would be nasty to simplify by hand…
1300
11
/ • 3∙5
/ = 3∙3 = 9
39 • 15 = 3∙13 • 3∙5 = 3∙13
/ 2∙5∙2 20
50 26
2∙5∙5 2∙13
2∙5∙5/ 2∙13
.
12
3
Math TLC (Math 010 and Math 110)
How to Solve Gateway Problems
Feb-09
Sample Problem #5 (continued)
Sample Problem #5: Dividing fractions
NOTE: You do NOT need an LCD when dividing fractions.
Step 1: Multiply the first fraction by the
reciprocal of the second fraction.
45 ÷ 21 = 45 • 26
13 26 14 21
(i.e. flip the second fraction upside down and change ÷ to • .)
Step 2: Factor both the numerators and denominators into
prime factors, then write each fraction in factored form:
First fraction:
45 = 3∙3∙5 and 13 = 13 (prime)
Second fraction:
26 = 2∙13 and 21 = 3∙7
So you can write 45 • 26 as 3∙3∙5 • 2∙13
13 21
13
3∙7
Step 3: Now just cancel any common factors that appear in
both numerator and denominator. Once you multiply out
any remaining factors, the result is your simplified answer.
/ • 2∙13
/ = 3∙5∙2 = 30
3∙3∙5
/
/3∙7
13
7
7
NOTE: Once again, it is much easier to factor first and then
cancel, rather than multiplying out the numerators and
denominators and then trying to simplify the answer (especially if
you aren’t using a calculator!) If you multiplied first, you’d have
gotten 1170 , which would be pretty hard to simplify by hand.
273
13
NOTE: Gateway problems 4 & 6
Full Solution to Sample Problem #5:
9
20
.
30
7
14
using mixed numbers both start
with the same step.
.
Here is the work we expect to see on your worksheet:
/ • 2∙13/
45 ÷ 21 = 45 • 26 = 3∙3∙5 • 2∙13 = 3∙3∙5
13 26 13 21
13
3∙7
13
/
3∙7
/
.
= 3∙5∙2 = 30
1∙7
7
15
A mixed number consists of an integer part
and a fraction part. We want to covert the
mixed number into an improper fraction,
This is done by multiplying the integer part
by the denominator of the fraction part,
then adding that product to the
numerator of the fraction and putting that
sum over the original denominator.
16
4
Math TLC (Math 010 and Math 110)
How to Solve Gateway Problems
Feb-09
Sample Problem #4: Multiplying mixed numbers
Sample Problem #4 (continued)
17
3
⋅ 76
Step 2: Factor both the numerators and denominators into
prime factors, then write each fraction in factored form:
Step 1: Convert the mixed number 5 23 into
an improper fraction: (Note that 5 23 = 5 + 23 = 15 + 32 )
.
5
1
+ 32 = 15⋅⋅33 + 32 = 153 + 32 = 173
17 and 3 are both prime
6 = 2∙3 and 7 is prime
So you can write 17 ∙ 6 as 17 ∙ 2∙3 .
3 7
3 7
Step 3: Now just cancel any common factors that appear in both
numerator and denominator. Once you multiply out any remaining
factors, the result is your simplified answer.
6 , which we can then
So 5 23 ⋅ 76 becomes 17
3 ⋅7
solve the same way we did problem #3.
17
/ = 17∙2 =
17 ∙ 2∙3
3/
7
7
34
7
.
18
Sample Problem #6: Dividing mixed numbers
Full Solution to Sample Problem #4:
Step 1: Convert the mixed numbers into improper
fractions:
Here is the work we expect to see on your worksheet:
5⋅3
15
17
2
2
5 23 = 5 + 23 = 15 + 23 = 13
⋅ + 3 = 3 + 3 = 3
5 23 ⋅ 76 = 173 ⋅ 76 = 173 ⋅ 27⋅3 = 17/3 ⋅ 27⋅3/ = 177⋅2 =
First fraction:
Second fraction:
34
7
19
7 17 = 7 + 17 = 17 + 17 =
7⋅7
1⋅7
+ 17 =
12 12 = 12 + 12 = 121 + 12 = 121⋅2⋅2 + 12 =
49
7
24
2
+ 17 =
50
7
+ 12 =
25
2
20
5
Math TLC (Math 010 and Math 110)
How to Solve Gateway Problems
Sample Problem #6 (continued) 7 17 ÷ 12 12 =
Feb-09
50
7
÷ 252 =
50
7
⋅ 252
Full Solution to Sample Problem #6:
Step 2: Factor both the numerators and denominators into
prime factors, then write each fraction in factored form:
First fraction:
50 = 2∙5∙5 and 7 is prime
Second fraction:
2 is prime and 25 = 5∙5
So you can write 50 • 2 as 2∙5∙5 • 2
7 25
7
5∙5
Here is the work we expect to see on your worksheet:
.
Step 3: Now just cancel any common factors that appear in
Both numerator and denominator. Once you multiply out any
remaining factors, the result is your simplified answer.
2∙5∙5
/ / • 2 = 2∙2 = 4
7
5∙5
7
7
/ /
21
NOTE: Gateway problems 7 & 8 both
7 17 = 7 + 17 = 71 + 71 =
7⋅7
1⋅7
+ 17 =
12 12 = 12 + 12 = 121 + 12 = 121⋅2⋅2 + 12 =
7 71 ÷ 12 12 =
=
2⋅5⋅5
7
⋅ 52⋅5 =
50
7
÷ 252 =
2⋅5/⋅5/
7
⋅ 5/2⋅5/ =
50
7
49
7
24
2
+ 17 =
+ 12 =
50
7
25
2
⋅ 252
2⋅2
7
=
4
7
22
Order of operations memory device:
require using the order of operations.
“Please excuse my dear Aunt Sally”
Order of operations:
1) First, calculate expressions within grouping symbols
(parentheses, brackets, braces,absolute values, fraction bars).
If there are nested sets of grouping symbols, start with the
innermost ones first and work your way out.
2) Exponential expressions – left to right
3) Multiplication and division – left to right
4) Addition and subtraction – left to right
1. Please
2. Excuse
3. My Dear
4. Aunt Sally
(Parentheses)
(Exponents)
(Multiply and Divide)
(Add and Subtract)
… or just remember
23
PEMDAS
24
6
Math TLC (Math 010 and Math 110)
How to Solve Gateway Problems
Feb-09
Sample Problem # 7: Order of Operations
Now calculate the bottom expression: 2(6+2) + 4
Step 1: Parentheses: 2(6+2) + 4 = 2(8) + 4
Step 2: Exponents: There aren’t any in this part.
Step 3: Multiply/Divide: 2(8) + 4 = 2•8 + 4 = 16 + 4
Strategy: Calculate out the entire top expression and then the entire bottom
expression, using the order of operations on each part. Then simplify the resulting
fraction, if necessary.
TOP EXPRESSION: 24 – 4(7 + 2)
Now put the top over the bottom and simplify the
resulting fraction:
Step 1: Parentheses:
24
Step 2: Exponents:
24 – 4(9) = 2•2•2•2 – 4(9) = 16 – 4(9)
– 4(7 + 2) =
24
Step 4: Add/Subtract: 16 + 4 = 20
– 4(9)
(because 2•2•2•2 = 4•2•2 = 8•2 = 16)
TOP = 24 – 4(7 + 2) = -20 = -1 = -1
BOTTOM
2(6+2) + 4
20
1
Step 3: Multiply/Divide: 16 – 4(9) = 16 – 4•9 = 16 – 36
Step 4: Add/Subtract:
16 – 36 = -20
25
26
Sample Problem # 8: Order of Operations
Full Solution to Sample Problem #7:
Here is the work we expect to see on your worksheet:
24 – 4(7 + 2) = 24 – 4(9) = 16 – 4(9) = 16 – 36 = -20 = -1 = -1
2(6+2) + 4
2(8) + 4
16 + 4
20
20 1
27
Strategy: Deal with the expressions inside the grouping
symbols (parentheses, brackets) first, starting with the
innermost set (-3 + 6).
STEP 1: (inside the parentheses)
3[17 + 5(-3 + 6) - 10] = 3[17 + 5(3) - 10]
STEP 2: (inside the brackets; multiply first, then add and subtract)
3[17 + 5(3) -10] = 3[17 + 5•3 -10] = 3[17 + 15 - 10]
= 3[17 + 15 - 10] = 3[32 - 10] = 3[22]
STEP 3: Do the final multiplication: 3[22] = 3•22 = 66 28
7
Math TLC (Math 010 and Math 110)
How to Solve Gateway Problems
Feb-09
Full Solution to Sample Problem #8:
Here is the work we expect to see on your worksheet:
3[17 + 5(-3 + 6) - 10] = 3[17 + 5(3) - 10] =
3[17 + 15 - 10] = 3[32 - 10] = 3[22] = 66
29
8