How to find the whole from the part - ISEE Middle Level Quantitative Reasoning
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Albert sent a letter to his penpal everyday for the entire month of January and February. How many letters did Albert send his penpal?
Albert sent a letter to his penpal everyday for the entire month of January and February. How many letters did Albert send his penpal?
Add the total days in January and February:
Answer: Albert sent his penpal 59 letters.
Add the total days in January and February:
Answer: Albert sent his penpal 59 letters.
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150% of what number is 9,000?
150% of what number is 9,000?
Taking 150% of a number is the same as multiplying that number by 1.5. We can find our number, therefore, by dividing 9,000 by 1.5:
Taking 150% of a number is the same as multiplying that number by 1.5. We can find our number, therefore, by dividing 9,000 by 1.5:
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is a multiple of 2, but not a multiple of 6. If 
is divided by 6, What could be the remainder?
If the remainder of dividing 
by 6 is 0, then 
must be a multiple of 6, which contradicts what we are given.
Suppose we call 
and 
the quotient and remainder, respectively, of dividing 
by 6. Then
If the remainder is 1, then
, as a product of two integers, one even, is an even number; if 1 is added, the result is odd. Therefore, if 1 is the remainder, 
is odd, which also contradicts what we are given about 
. 1 cannot be the remainder, and for similar reasons, neither can 3.
This leaves 2, which can be the remainder; for example, 8, which is a multiple of 2, but not a multiple of 6, yields a quotient of 1 and a remainder of 2 when divided by 2.
If the remainder of dividing
Suppose we call
If the remainder is 1, then
This leaves 2, which can be the remainder; for example, 8, which is a multiple of 2, but not a multiple of 6, yields a quotient of 1 and a remainder of 2 when divided by 2.
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After a discount of 20%, a boom box costs \$144.
Which is the greater quantity?
(a) The price of the boom box before the discount
(b) \$172.80
After a discount of 20%, a boom box costs \$144.
Which is the greater quantity?
(a) The price of the boom box before the discount
(b) \$172.80
Paying for something at 20% discount is the same as paying for it at 80% of the price.
If the boom box costs \$172.80 before a 20% discount, then afterward, it costs 80% of this, or
Therefore, the original price must have been greater than \$172.80.
Paying for something at 20% discount is the same as paying for it at 80% of the price.
If the boom box costs \$172.80 before a 20% discount, then afterward, it costs 80% of this, or
Therefore, the original price must have been greater than \$172.80.
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40% of 
is 200
70% of 
is 350
Which is the greater quantity?
(a) 
(b) 
40% of
70% of
Which is the greater quantity?
(a)
(b)
Each statement can be written as a proportion, and solved for its variable by cross-multiplying. In each case, we replace as follows:
(a) The percent is 40, the part is 200, and the whole is 
:
(b) The percent is 70, the part is 350, and the whole is 
:
Each statement can be written as a proportion, and solved for its variable by cross-multiplying. In each case, we replace as follows:
(a) The percent is 40, the part is 200, and the whole is
(b) The percent is 70, the part is 350, and the whole is
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Chet bought an MP3 player and got \$14.50 back in change. The tax on the MP3 player was 5%.
Which is the greater quantity?
(a) The price of the MP3 player before tax
(b) 
Chet bought an MP3 player and got \$14.50 back in change. The tax on the MP3 player was 5%.
Which is the greater quantity?
(a) The price of the MP3 player before tax
(b)
No clue is given as to how much Chet paid for the device - neither the price paid for the player nor the amount of money Chet gave the clerk is given here.
No clue is given as to how much Chet paid for the device - neither the price paid for the player nor the amount of money Chet gave the clerk is given here.
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60% of 3,000 is equal to 40% of what number?
60% of 3,000 is equal to 40% of what number?
60% of 3,000 is equal to 
.
40% of an unknown number 
can be written as 
. Set these equal and solve for 
:
60% of 3,000 is equal to
40% of an unknown number
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Claude bought a new game for \$32 at 20% off its original price. What was the original price of the game?
Claude bought a new game for \$32 at 20% off its original price. What was the original price of the game?
Since the game was 20% off, that means the game's value is currently 80% of the original price.
In order to figure out what the original price of the game was, you must set up a proportion:
The proportion is set up this way because the \$32 Claude paid is equal to 80% of the price of the game, not 100%. The x stands for the unknown original price of the game. Next, cross multiply. The bottom number of each fraction should be multiplied by the top number.
The above will be your result. You then must solve for x. To do this, divide by 80 on each side of the equation.
The value of x is your answer.
Since the game was 20% off, that means the game's value is currently 80% of the original price.
In order to figure out what the original price of the game was, you must set up a proportion:
The proportion is set up this way because the \$32 Claude paid is equal to 80% of the price of the game, not 100%. The x stands for the unknown original price of the game. Next, cross multiply. The bottom number of each fraction should be multiplied by the top number.
The above will be your result. You then must solve for x. To do this, divide by 80 on each side of the equation.
The value of x is your answer.
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is 
% of what number?
Round to the nearest thousandth.
Round to the nearest thousandth.
Remember that for percentages, the key to setting up the problem is intelligent translation. The word "is" becomes 
, "of" signals multiplication, "what" (and equivalent words) signal a variable (
).
Therefore, we can translate:
is 
% of what number?
As...
To solve, divide both sides by 
:
Rounded, this is:
Remember that for percentages, the key to setting up the problem is intelligent translation. The word "is" becomes
Therefore, we can translate:
As...
To solve, divide both sides by
Rounded, this is:
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is 
% of what number?
Round to the nearest thousandth.
Round to the nearest thousandth.
Remember that for percentages, the key to setting up the problem is intelligent translation. The word "is" becomes 
, "of" signals multiplication, "what" (and equivalent words) signal a variable (
).
Therefore, we can translate:
is 
% of what number?
As . . .
To solve, divide both sides by 
:
Rounded, this is:
Remember that for percentages, the key to setting up the problem is intelligent translation. The word "is" becomes
Therefore, we can translate:
As . . .
To solve, divide both sides by
Rounded, this is:
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is 
% of what number?
Round to the nearest thousandth.
Round to the nearest thousandth.
Remember that for percentages, the key to setting up the problem is intelligent translation. The word "is" becomes 
, "of" signals multiplication, "what" (and equivalent words) signal a variable (
).
Therefore, we can translate:
is 
% of what number?
As...
To solve, divide both sides by 
:
Rounded, this is:
Remember that for percentages, the key to setting up the problem is intelligent translation. The word "is" becomes
Therefore, we can translate:
As...
To solve, divide both sides by
Rounded, this is:
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500 is 
of what number?
500 is
of a number is equal to
of the number.
The question becomes equivalent to asking this - 500 is 
of what number?
We can find out by dividing 500 by 
:
.
of the number.
The question becomes equivalent to asking this - 500 is
We can find out by dividing 500 by
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A can of tomato soup cost $.85. How much does it cost to buy 12 cans of soup?
A can of tomato soup cost $.85. How much does it cost to buy 12 cans of soup?
Multiply:
Multiply:
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Three pounds of flour costs \$2.50. How much does 9 pounds costs?
Three pounds of flour costs \$2.50. How much does 9 pounds costs?
Multiply:
Multiply:
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Each day Gillian crosses the bridge, she has to pay a \$6.50 toll. How much does she spend on tolls in one week?
Each day Gillian crosses the bridge, she has to pay a \$6.50 toll. How much does she spend on tolls in one week?
Multiply:
Multiply:
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Mary went shopping for her band uniform. She spent \$29.00 on pants, \$25.42 on a shirt, and \$25.42 on shoes. How much did Mary spend in total?
Mary went shopping for her band uniform. She spent \$29.00 on pants, \$25.42 on a shirt, and \$25.42 on shoes. How much did Mary spend in total?
Add up all of Mary's expenses:
Add up all of Mary's expenses:
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Multiply: 
Multiply:
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There are 
pieces of candy for each of the 
children at a birthday party. How many pieces of candy are there all together?
There are
Multiply: 
pieces of candy
Multiply:
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Calvin owns a restaurant. He pays \$1,200.00 a month in rent. How much rent does Calvin pay a year?
Calvin owns a restaurant. He pays \$1,200.00 a month in rent. How much rent does Calvin pay a year?
Multiply:
Answer: Calvin pays \$14,400.00 in rent a year.
Multiply:
Answer: Calvin pays \$14,400.00 in rent a year.
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Multiply:
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