ISEE Upper Level

Decimal Operations

Arithmetic with decimals and understanding place value

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Imagine you are buying snacks at a movie theater or upgrading your character in a video game. You need to know exactly how much money you have! Decimals are basically just money math, and mastering them is like having a superpower for the ISEE. 🍿🎮

The ISEE loves to test if you know exactly where the little dot goes. In the Mathematics Achievement section, you will often see answer choices with the exact same digits, but the decimal point is in different spots! So, how do we handle these dots? It depends on the operation.

When adding or subtracting decimals, you must line up the dots! Think of it like buttoning a shirt—if you don't line up the buttons and holes, everything gets messy. Add some invisible zero placeholders if you need to!

When multiplying decimals, ignore the dots at first. Pretend they aren't even there! Multiply the numbers like normal, then count up all the decimal places in the original problem. That total is exactly how many decimal places your answer needs.

Finally, when dividing, nobody likes dividing by a decimal. Move the dot in the outside number to make it a whole number, and do the exact same move for the inside number. Remember, there is no penalty for guessing on the ISEE. If a decimal problem looks scary, try estimating the answer first. You've got this! 🚀

Key Formula

When multiplying decimals, the total number of decimal places in your answer equals the sum of the decimal places in the numbers you are multiplying. If

A

has 2 decimal places and

B

has 1 decimal place,

A \times B

will have

2 + 1 = 3

decimal places.

Practice Questions

4 practice questions for ISEE Upper Level

Q1 Hard

Evaluate

(1.25 \times 0.8) + (13.2 \div 0.11) - 5.05

116.95

115.95

117.05

116.05

Show Solution

Step 1: Calculate the first term $1.25 \times 0.8$ .

$1.25 \times 0.8 = 1.000 = 1$
Step 2: Calculate the second term $13.2 \div 0.11$ .

To divide by a decimal, we can multiply both numbers by a power of 10 to make the divisor a whole number. Multiply both by $100$ :
$13.2 \times 100 = 1320$
$0.11 \times 100 = 11$
So, $13.2 \div 0.11 = 1320 \div 11 = 120$
Step 3: Substitute these values back into the expression:

$1 + 120 - 5.05$
Step 4: Perform the addition and subtraction:

$121 - 5.05 = 115.95$

Answer: B

Q2 Hard

A sculptor purchases

3.5

kilograms of bronze at $

12.80

per kilogram, and

2.25

kilograms of copper at $

9.60

per kilogram. She also buys a special adhesive for $

18.75

. If she pays with a $

100

bill, how much change should she receive?

A $

15.15

B $

14.85

C $

13.95

D $

16.05

Show Solution

Step 1: Calculate the cost of the bronze.

Cost of bronze = $3.5 kg \times 12.80 per kg = 44.80$
Step 2: Calculate the cost of the copper.

Cost of copper = $2.25 kg \times 9.60 per kg = 21.60$
Step 3: Calculate the total cost of all items.

Total cost = Cost of bronze + Cost of copper + Cost of adhesive
Total cost = $44.80 + 21.60 + 18.75 = 85.15$
Step 4: Calculate the change received.

Change = Amount paid - Total cost
Change = $100 - 85.15 = 14.85$

Answer: B

Q3 Hard

Evaluate

(0.006 \times 5000) - (2.4 \div 0.08) + (0.1 \times 0.05)

0.005

0.5

30.005

- 0.005

Show Solution

Step 1: Calculate the first term $0.006 \times 5000$ .

$0.006 \times 5000 = 6 \times 5 = 30$ (You can think of $0.006 \times 1000 = 6$ , then $6 \times 5 = 30$ ).
Step 2: Calculate the second term $2.4 \div 0.08$ .

To divide by a decimal, multiply both numbers by a power of 10 to make the divisor a whole number. Multiply both by $100$ :
$2.4 \times 100 = 240$
$0.08 \times 100 = 8$
So, $2.4 \div 0.08 = 240 \div 8 = 30$
Step 3: Calculate the third term $0.1 \times 0.05$ .

$0.1 \times 0.05 = 0.005$ (Multiply $1 \times 5 = 5$ . There is one decimal place in $0.1$ and two in $0.05$ , so three decimal places in the product).
Step 4: Substitute these values back into the expression:

$30 - 30 + 0.005$
Step 5: Perform the subtraction and addition:

$0 + 0.005 = 0.005$

Answer: A

Q4 Hard

A family plans a

480

-mile road trip. Their car gets an average of

32

miles per gallon. If gasoline costs $

3.15

per gallon, and they estimate spending an additional $

7.50

on tolls, what is their estimated total cost for fuel and tolls for the trip?

A $

54.75

B $

47.25

C $

55.25

D $

62.25

Show Solution

Step 1: Calculate the total number of gallons of gasoline needed for the trip.

Gallons needed = Total miles $\div$ Miles per gallon
Gallons needed = $480 \div 32 = 15$ gallons
Step 2: Calculate the total cost of the gasoline.

Cost of gasoline = Gallons needed $\times$ Cost per gallon
Cost of gasoline = $15 \times 3.15 = 47.25$
Step 3: Calculate the estimated total cost for fuel and tolls.

Total cost = Cost of gasoline + Cost of tolls
Total cost = $47.25 + 7.50 = 54.75$

Answer: A

Tips & Strategies

Use estimation to cheat! If you see $3.1 \times 4.8$ , just think $3 \times 5 = 15$ . The ISEE answer choices might be $1.488$ , $14.88$ , $148.8$ , and $1488$ . You know the answer must be close to $15$ , so pick $14.88$ instantly without doing the hard math!
When adding or subtracting whole numbers and decimals together (like $5 - 2.34$ ), always add placeholder zeros to the whole number first ( $5.00 - 2.34$ ). It prevents silly borrowing mistakes.

Common Mistakes

Watch out for lining up the decimals when multiplying! You only line up decimals for adding and subtracting. For multiplying, just line up the numbers on the right side.
Don't forget to move the decimal point in BOTH numbers when dividing. If you move the decimal one spot in the outside number (divisor), you MUST move it one spot in the inside number (dividend) too!

Frequently Asked Questions

How do I know if I should add, subtract, multiply, or divide on decimal word problems?

Look for keywords! Words like 'per', 'each', or 'rate' usually mean you need to multiply or divide. Words like 'total', 'combined', or 'difference' mean you should add or subtract.

What happens if I run out of numbers when moving the decimal point to the left?

Just add zeros! Think of the zeros as empty lily pads for your decimal point to jump onto. For example, moving the decimal 3 places left on the number $5$ becomes $0.005$ .

Is there a penalty for guessing if I get stuck on a hard decimal problem?

Nope! The ISEE does not penalize you for wrong answers. If the decimal math is taking too long, use estimation to cross out the crazy answers, and then guess from the remaining choices.

Are decimals and fractions related on the ISEE?

Yes, they are math twins wearing different outfits! For example, $0.5$ is exactly the same as $\frac{1}{2}$ . Sometimes, converting a tricky decimal into a fraction makes the problem much easier to solve.

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