PICAT EXAM STUDY GUIDE: ARITHMETIC
REASONING #1 – 50 Q’S AND A’S
Karen is 6 years older than her sister Michelle, and Michelle is 2 years
younger than their brother David. If the sum of their ages is 62, how old is
Michelle? - -Answer: 18
Let: x = Michelle's age
x + 6 = Karen's age
x + 2 = David's age
State the problem in an equation:
x + x + 6 + x + 2 = 62
3x + 8 = 62
3x + 8 - 8 = 62 - 8
3x = 54
3x /(the slash means divide) 3 =
x = 18
This is Michelle's age.
- A barista averages making 16 coffees per hour. At this rate, how many
hours will it take until she's made 1,200 coffees? - -Answer: 75 hours
To solve the problem, simply set up a proportion using what you know and
using x for what you don't know:
16 coffees / 1 hour = 1200 coffees / x hours
Cross-multiply, and you have:
16x = 1200
Divide both sides by 16 to isolate x and you have:
x = 75
You may be able to set up this problem in your head by just dividing 1200 by
16. You will get the same answer, but it's important to know how to use a
proportion since these problems are not always this simple. Using a
proportion also helps you to reason why you need to do the final operation
and this will help on some of the newer tests where you may be asked the
reason for an answer.
- Camille uses a 20% off coupon when buying a sweater that costs $47.99. If
she also pays 6% sales tax on the purchase, how much does she pay? - -
Answer: $40.69
First, compute the price of the sweater after the discount:
Discounted price of the sweater:
(100% - 20%)($47.99) = 80%($47.99)
.80(47.99) = $38.39
Second, add the sales tax:
Total amount paid:
$38.39 + $38.39(6%)
,38.39 + 38.39(6%)
38.39 + 2.30 = $40.69
- At Michael's smoothie shop smoothies cost $3.50 during the week and
$4.00 on weekends. If Jonathan bought a smoothie every day except
Saturday how much did he spend on smoothies that week? - -Answer:
$21.50
Compute the total amount paid for smoothies bought from Monday to Friday
(5 days) at $3.50 / smoothie:
5 x $3.50 = $17.50
Add this to the amount paid for the smoothie bought on Sunday at $4.00 /
smoothie:
$17.50 + $4.00 = $21.50
- In a classroom of 32 students, 14 are male. What percentage of the class is
female? - -Answer: 56%
First, solve how many females are in the class:
Number of females = 32 − 14 = 18
To compute for the percentage of females in the class:
Percentage = ()(100%) = 56.25% = 56%
- Cory's average score after 8 tests is 91%. If he gets a 78% on his 9th test,
what will his new average be? - -Answer: 89.6%
The formula for average is:
sum of all values given / number of all values given
For the 8 tests, the average is presented as 91. While we don't know the
exact score on each test, you have to use the information that is given. Since
the average of all previous scores is 91, assign the score of 91 to each of the
8 previous tests.
This gives us 8 tests with a score of 91 each.
91 x 8 = 728
To get new average after the 9th test, use the same formula:
728 + = = 89.5
Since there are 2 answer choices that are very close to this, and you don't
know which way the .5 will be rounded, extend the division another place by
adding a 0 to the dividend, then divide.
You find that the answer is now 89.55, which rounds up to 89.6. Therefore,
you now know that 89.5 would not be correct. Always extend the decimal
places when 2 answer choices are as close as these and you don't know
which one is correct.
89.6%
- Julie gives 7 pieces of candy to each of her friends. If Julie gives all of her
candy away, which amount of candy could have been the amount she
distributed? - -Answer: 161
, The problem states that all of Julie's friends received 7 pieces each. There is
no mention about the number of friends Julie gave the candies to. But no
matter how many friends Julie gave the candies to, the total number of
candies she has given away must be divisible by 7. There is no other way to
solve the problem but try each given choice and pick one that is divisible by
7 and results in a whole number.
= 23
= 18.28
= 20.71
= 23.14
Therefore, the answer is 161 candies (distributed to 23 friends with 7 candies
each).
- Four out of 28 students had to go to summer school. What is the ratio of
students who did not have to go to summer school expressed, in its lowest
terms? - -Answer:
First of all, remember that you are solving for the number of students who
did not have to go to summer school. If 4 students did have to go, out of 28,
then 24 students did not have to go.
28 - 4 = 24
To express this as a ratio, simply treat 24 as the numerator and 28 as
denominator:
divide by =
Divide both numerator and denominator by 4 to reduce the ratio or fraction
to the lowest terms.
TIP: When reducing a fraction to its lowest terms, choose the biggest
common number that can be used to divide both numbers. In this case, it is
4 because 24 can be divided by 4, and 28 can be divided by 4.
Also, remember that ratios can be expressed in three ways:
in words - 6 to 7
using a colon - 6:7
and as a fraction -
These are all equivalent forms of this ratio. Sometimes it's easier to work
with ratios when expressed as fractions.
- If doughnuts are usually 75 cents each, but there is a sale on Friday
advertising them as 1 dozen + ½ dozen for $10.80, what is the new cost for
just one doughnut? - -Answer: 60 cents
Compute how many doughnuts are there in 1 + dozen:
Remember, 1 dozen = 12 doughnuts.
12 + x 12 = 12 + 6 = 18 doughnuts
If 18 doughnuts cost $10.80, compute for the cost of each doughnut:
$10. = $0.60 per doughnut
The answer, therefore, is $0.60 or 60 cents.
REASONING #1 – 50 Q’S AND A’S
Karen is 6 years older than her sister Michelle, and Michelle is 2 years
younger than their brother David. If the sum of their ages is 62, how old is
Michelle? - -Answer: 18
Let: x = Michelle's age
x + 6 = Karen's age
x + 2 = David's age
State the problem in an equation:
x + x + 6 + x + 2 = 62
3x + 8 = 62
3x + 8 - 8 = 62 - 8
3x = 54
3x /(the slash means divide) 3 =
x = 18
This is Michelle's age.
- A barista averages making 16 coffees per hour. At this rate, how many
hours will it take until she's made 1,200 coffees? - -Answer: 75 hours
To solve the problem, simply set up a proportion using what you know and
using x for what you don't know:
16 coffees / 1 hour = 1200 coffees / x hours
Cross-multiply, and you have:
16x = 1200
Divide both sides by 16 to isolate x and you have:
x = 75
You may be able to set up this problem in your head by just dividing 1200 by
16. You will get the same answer, but it's important to know how to use a
proportion since these problems are not always this simple. Using a
proportion also helps you to reason why you need to do the final operation
and this will help on some of the newer tests where you may be asked the
reason for an answer.
- Camille uses a 20% off coupon when buying a sweater that costs $47.99. If
she also pays 6% sales tax on the purchase, how much does she pay? - -
Answer: $40.69
First, compute the price of the sweater after the discount:
Discounted price of the sweater:
(100% - 20%)($47.99) = 80%($47.99)
.80(47.99) = $38.39
Second, add the sales tax:
Total amount paid:
$38.39 + $38.39(6%)
,38.39 + 38.39(6%)
38.39 + 2.30 = $40.69
- At Michael's smoothie shop smoothies cost $3.50 during the week and
$4.00 on weekends. If Jonathan bought a smoothie every day except
Saturday how much did he spend on smoothies that week? - -Answer:
$21.50
Compute the total amount paid for smoothies bought from Monday to Friday
(5 days) at $3.50 / smoothie:
5 x $3.50 = $17.50
Add this to the amount paid for the smoothie bought on Sunday at $4.00 /
smoothie:
$17.50 + $4.00 = $21.50
- In a classroom of 32 students, 14 are male. What percentage of the class is
female? - -Answer: 56%
First, solve how many females are in the class:
Number of females = 32 − 14 = 18
To compute for the percentage of females in the class:
Percentage = ()(100%) = 56.25% = 56%
- Cory's average score after 8 tests is 91%. If he gets a 78% on his 9th test,
what will his new average be? - -Answer: 89.6%
The formula for average is:
sum of all values given / number of all values given
For the 8 tests, the average is presented as 91. While we don't know the
exact score on each test, you have to use the information that is given. Since
the average of all previous scores is 91, assign the score of 91 to each of the
8 previous tests.
This gives us 8 tests with a score of 91 each.
91 x 8 = 728
To get new average after the 9th test, use the same formula:
728 + = = 89.5
Since there are 2 answer choices that are very close to this, and you don't
know which way the .5 will be rounded, extend the division another place by
adding a 0 to the dividend, then divide.
You find that the answer is now 89.55, which rounds up to 89.6. Therefore,
you now know that 89.5 would not be correct. Always extend the decimal
places when 2 answer choices are as close as these and you don't know
which one is correct.
89.6%
- Julie gives 7 pieces of candy to each of her friends. If Julie gives all of her
candy away, which amount of candy could have been the amount she
distributed? - -Answer: 161
, The problem states that all of Julie's friends received 7 pieces each. There is
no mention about the number of friends Julie gave the candies to. But no
matter how many friends Julie gave the candies to, the total number of
candies she has given away must be divisible by 7. There is no other way to
solve the problem but try each given choice and pick one that is divisible by
7 and results in a whole number.
= 23
= 18.28
= 20.71
= 23.14
Therefore, the answer is 161 candies (distributed to 23 friends with 7 candies
each).
- Four out of 28 students had to go to summer school. What is the ratio of
students who did not have to go to summer school expressed, in its lowest
terms? - -Answer:
First of all, remember that you are solving for the number of students who
did not have to go to summer school. If 4 students did have to go, out of 28,
then 24 students did not have to go.
28 - 4 = 24
To express this as a ratio, simply treat 24 as the numerator and 28 as
denominator:
divide by =
Divide both numerator and denominator by 4 to reduce the ratio or fraction
to the lowest terms.
TIP: When reducing a fraction to its lowest terms, choose the biggest
common number that can be used to divide both numbers. In this case, it is
4 because 24 can be divided by 4, and 28 can be divided by 4.
Also, remember that ratios can be expressed in three ways:
in words - 6 to 7
using a colon - 6:7
and as a fraction -
These are all equivalent forms of this ratio. Sometimes it's easier to work
with ratios when expressed as fractions.
- If doughnuts are usually 75 cents each, but there is a sale on Friday
advertising them as 1 dozen + ½ dozen for $10.80, what is the new cost for
just one doughnut? - -Answer: 60 cents
Compute how many doughnuts are there in 1 + dozen:
Remember, 1 dozen = 12 doughnuts.
12 + x 12 = 12 + 6 = 18 doughnuts
If 18 doughnuts cost $10.80, compute for the cost of each doughnut:
$10. = $0.60 per doughnut
The answer, therefore, is $0.60 or 60 cents.
