Sample Gateway Problems: . . Working with Fractions and the Order of Operations Without Using a Calculator

1
Sample Gateway Problems..Working with
Fractions and the Order of OperationsWithout
Using a Calculator
2
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.)

3
Sample Problem 1 Adding fractions
Step 1 Factor the two denominators into prime
factors, then write each fraction with
its denominator in factored form 10 25
and 35 57, so 3 2 3 2
. .
10 35 25 57
Step 2 Find the least common denominator
(LCD) LCD 257
4
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.
LCD 257 37 2 2
. 257
572
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 572 257 by 257 572
257 257 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 55 55 5
5 . 257 257
257 27 14
/ /
5
Full Solution to Sample Problem 1
5 14
Here is the work we expect to see on your
worksheet 10 25 and 35 57, so
3 2 3 2 , and LCD 257

10 35 25 57 3 2 37
2 2 21 4 25 55
55 5 5 25 57 257
572 257 257 257 257
257 27 14
/ /
6
Sample Problem 2 Subtracting fractions
Step 1 Factor the two denominators into prime
factors, then write each fraction with
its denominator in factored form 14 27
and 35 57, so 5 - 2
.

27 57
Step 2 Find the least common denominator
(LCD) LCD 275
7
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
275 55 - 2 2
275 572
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 .
257 572 257
257
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 37 37
3 3 .
257 257 257 25 10
/ /
8
Full Solution to Sample Problem 2
3 . 10
Here is the work we expect to see on your
worksheet 14 27 and 35 57, so
5 - 2 5 - 2 , and LCD
257
14 35 27 57 5 - 2
55 - 2 2 25 - 4 21
37 37 3 3 27 57
275 572 257 257 257
257 257 25 10
/ /
9
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.)

10
Sample Problem 3 Multiplying fractions
Step 1 Factor both the numerators and
denominators into prime factors, then write each
fraction in factored form First fraction
39 313 and 50 255 Second fraction
15 35 and 26 213 So you can write
39 15 as 313 35 .

50 26 255 213 NOTE You
do NOT need an LCD when multiplying
fractions.
11
Sample Problem 3 (continued)
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.

313 35 33 9 .
255 213
252 20
/ / / /
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 arent
using a calculator!) If you multiplied first,
youd have gotten 585 , which would be
nasty to simplify by hand 1300
12
Full Solution to Sample Problem 3
9 . 20
Here is the work we expect to see on your
worksheet 39 15 313 35
313 35 33 9 .
50 26 255 213
255 213 252 20
/ / / /
13
Sample Problem 5 Dividing fractions
Step 1 Multiply the first fraction by the
45 21 45 26 reciprocal of the second
fraction.
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 45 335 and 13 13
(prime) Second fraction 26 213 and
21 37 So you can write 45 26
as 335 213 .

13 21 13 37
14
Sample Problem 5 (continued)
NOTE You do NOT need an LCD when dividing
fractions. 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.

335 213 352 30
13 37 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
arent using a calculator!) If you multiplied
first, youd have gotten 1170 , which would be
pretty hard to simplify by hand.
273
15
Full Solution to Sample Problem 5
30 . 7
9 . 20
Here is the work we expect to see on your
worksheet 45 21 45 26 335 213
335 213 13 26
13 21 13 37 13
37 . 352 30 17 7
/ / / /
16
NOTE Gateway problems 4 6 using mixed
numbers both start with the same step.

• 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.

17
Sample Problem 4 Multiplying mixed numbers
Step 1 Convert the mixed number into
an improper fraction (Note that
) .


So becomes , which we can
then solve the same way we did problem 3.
18
Sample Problem 4 (continued)
Step 2 Factor both the numerators and
denominators into prime factors, then write each
fraction in factored form First fraction
17 and 3 are both prime Second fraction
6 23 and 7 is prime So you can write 17
6 as 17 23 .
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.
/ /
17 23 172 34 .
3 7
7 7
19
Full Solution to Sample Problem 4
Here is the work we expect to see on your
worksheet
/ /
20
Sample Problem 6 Dividing mixed numbers
Step 1 Convert the mixed numbers into improper
fractions


21
Sample Problem 6 (continued)
Step 2 Factor both the numerators and
denominators into prime factors, then write each
fraction in factored form First fraction
50 255 and 7 is prime Second
fraction 2 is prime and 25 55 So you can
write 50 2 as 255 2 .

7 25 7 55 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. 255 2
22 4 7
55 7 7
/ / / /
22
Full Solution to Sample Problem 6
Here is the work we expect to see on your
worksheet
/ / / /
23
NOTE Gateway problems 7 8 both require using
the order of operations.

• Order of operations
• 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.
• Exponential expressions left to right
• Multiplication and division left to right
• Addition and subtraction left to right

24

• Order of operations memory device
• Please excuse my dear Aunt Sally
• 1. Please (Parentheses)
• 2. Excuse (Exponents)
• 3. My Dear (Multiply and Divide)
• 4. Aunt Sally (Add and Subtract)
• or just remember PEMDAS

25
Sample Problem 7 Order of Operations
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) Step 1
Parentheses 24 4(7 2) 24 4(9)
Step 2 Exponents 24 4(9) 2222
4(9) 16 4(9)
(because 2222 422 82
16) Step 3 Multiply/Divide 16 4(9) 16
49 16 36 Step 4 Add/Subtract 16
36 -20
26
Now calculate the bottom expression 2(62) 4

• Step 1 Parentheses 2(62) 4 2(8) 4
• Step 2 Exponents There arent any in this part.
• Step 3 Multiply/Divide 2(8) 4 28 4
  16 4
• Step 4 Add/Subtract 16 4 20
• Now put the top over the bottom and simplify the
  resulting fraction
• TOP 24 4(7 2) -20 -1
  -1
• BOTTOM 2(62) 4 20 1

27
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(62) 4 2(8)
4 16 4 20 20
1
28
Sample Problem 8 Order of Operations

• Strategy Deal with the expressions inside the
  grouping symbols (parentheses, brackets) first,
  starting with the innermost set (-3 6).
• STEP 1 (inside the parentheses)
• 317 5(-3 6) - 10 317
  5(3) - 10
• STEP 2 (inside the brackets multiply first,
  then add and subtract)
• 317 5(3) -10 317 53
  -10 317 15 - 10
• 317 15 - 10 332 - 10
  322
• STEP 3 Do the final multiplication 322 322
  66

29
Full Solution to Sample Problem 8
Here is the work we expect to see on your
worksheet
317 5(-3 6) - 10 317 5(3) - 10
317 15 - 10 332 - 10 322 66