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Problem Bank

For every problem, there is one solution which is simple, neat, and wrong.- H. L. Mencken

This page is under construction.

Most of these problems on this page have three images to click on. You can see:

a video step-by-step answer

a written step-by-step answer

only the answer

Apologies if those icons look slightly large on a computer. This page is optimized to look and work great on a phone held horizontally, an on a tablet.

It is up to you to use the problem bank wisely. Use some of the problems for step-by-step guidance as you learn to mimic an algorithm. Use other problems as practice with frequent hints as needed. Use other problems to test yourself to make sure you are not falling behind as the term progresses.

Build the habit of asking yourself if each answer you get is reasonable. Check your answers so you do not practice bad habits.

Give the place value of each digit in 305,964.
Name each digit one at a time.

3 hundred thousands

0 ten thousands

5 thousands

9 hundreds

6 tens

4 ones
3 hundred thousands, 0 ten thousands, 5 thousands, 9 hundreds, 6 tens, 4 ones

Give the place value of each digit in 73,890,672,540.
Name each digit one at a time.

7 ten billions

3 billions

8 hundred millions

9 ten millions

0 millions

6 hundred thousands

7 ten thousands

2 thousands

5 hundreds

4 tens

0 ones
7 ten billions, 3 billions, 8 hundred millions, 9 ten millions, 0 millions, 6 hundred thousands, 7 ten thousands, 2 thousands, 5 hundreds, 4 tens, 0 ones

In the number 546,789, which digit tells the number of hundred thousands?
Identify the requested place value: **5**,46,789

So the 5 does
the 5 does

Round 5,3825,382 to the nearest hundred.
Identify the requested place value: 5,3825,**3**82

Also consider the place value immediately to the right: 5,3825,3**8**2.

Is the 8 as big as 5 or more? Yes, so it makes us increase the 3 before we zero out that 8 and everything to its right: 5,3825,400
5,3825,400

Round 9,494 to the nearest ten.
Identify the requested place value: 9,4**9**4

Also consider the place value immediately to the right: 9,49**4**.

Is the 4 as big as 5 or more? No, so leave the 9 unchanged as we zero out that 4 (there is nothing to its right): 9,490
9,490

Round 973,973 to the nearest hundred.
Identify the requested place value: 973,**9**73

Also consider the place value immediately to the right: 973,9**7**3.

Is the 7 as big as 5 or more? Yes, so it makes us increase the 9 before we zero out that 7 and everything to its right.

Increasing the 9 causes carrying: 974,000
974,000

Round 47,256,344 to the nearest million.
Identify the requested place value: 4**7**,256,334

Also consider the place value immediately to the right: 47,**2**56,334

Is the 2 as big as 5 or more? No, so leave the 7 unchanged as we zero out that 2 and everything to its right: 47,000,000
47,000,000

Round 34,528 to the nearest thousand.
Identify the requested place value: 3**4**,527

Also consider the place value immediately to the right: 34,**5**28

Is the 5 as big as 5 or more? Yes, so it makes us increase the 4 before we zero out that 5 and everything to its right: 35,000
35,000

Round 34,528 to the nearest ten.
Identify the requested place value: 34,5**2**8

Also consider the place value immediately to the right: 34,52**8**

Is the 8 as big as 5 or more? Yes, so it makes us increase the 2 before we zero out that 8 (there is nothing to its right): 35,530
35,530

Round 34,528 to the nearest hundred.
Identify the requested place value: 34,**5**28

Also consider the place value immediately to the right: 34,5**2**8

Is the 2 as big as 5 or more? No, so leave the 5 unchanged as we zero out that 2 and everything to its right: 34,500
34,500

Round 5.6783 to the nearest one.
Identify the requested place value: **5**.6783

Also consider the place value immediately to the right: 5.**6**783

Is the 6 as big as 5 or more? Yes, so it makes us increase the 5 before we zero out that 6 and everything to its right: 6
6

Round 5.6783 to the nearest hundredth.
Identify the requested place value: 5.6**7**83

Also consider the place value immediately to the right: 5.67**8**3

Is the 8 as big as 5 or more? Yes, so it makes us increase the 7 before we zero out that 8 and everything to its right: 5.68
5.68

Round 5.6783 to the nearest thousandth.
Identify the requested place value: 5.67**8**3

Also consider the place value immediately to the right: 5.678**3**

Is the 3 as big as 5 or more? No, so leave the 8 unchanged as we zero out that 3 (there is nothing to its right): 5.678
5.678

Round 5.6783 to the nearest tenth.
Identify the requested place value: 5.**6**783

Also consider the place value immediately to the right: 5.6**7**73

Is the 7 as big as 5 or more? Yes, so it makes us increase the 6 before we zero out that 7 and everything to its right: 5.7
5.7

Estimate 4,872 + 1,691 + 777 + 6,124 by first rounding each number to the nearest thousand.
≈ 5,000 + 2,000 + 1,000 + 6,000 = **14,000**
14,000

Estimate the sum 23,649 + 54,746 by first rounding to the nearest hundred.
≈ 23,600 + 54,700 = **78,300**
78,300

Estimate the difference 54,751 − 23,649 by first rounding to the nearest hundred.
≈ 54,800 − 23,600 = **31,200**
31,200

Estimate the product 824 × 489 by first rounding to the nearest hundred.
≈ 800 × 500 = **400,000**
400,000

Estimate the product 8.91 × 22.457 by first rounding to the nearest one.
≈ 9 × 22 = **198**
198

Estimate the quotient 78.2209 ÷ 16.09 by first rounding to the nearest ten.
≈ 80 ÷ 20 = **4**
4

(we need some problems here)

Identify the products and factors: 30 = 2 × 3 × 5
The answer to a multiplication problem is the **product**.

The amounts being multiplied are the **factors**.

So the product is 30, and the factors are 2, 3, and 5.
The product is 30. The factors are 2, 3, and 5.

Identify the products and factors: 9 × 8 = 72
The answer to a multiplication problem is the **product**.

The amounts being multiplied are the **factors**.

So the product is 72, and the factors are 8 and 9.
The product is 72. The factors are 8 and 9.

Determine whether 784 is divisible by 9.
The sum of the digits is 7 + 8 + 4 = 19.

Does 9 go into 19? No.

So 9 does not go into our original number either.
No.

Determine whether 5,552 is divisible by 5.
The one's place value digit is 5

Is this 0 or 5? Yes.

So 5 goes into our number.
Yes.

Determine whether 2,322 is divisible by 6.
The one's place value digit is 2

Is this 0, 2, 4, 6, or 8? Yes.

So 2 goes into our number.

The sum of the digits is 2 + 3 + 2 + 2 = 9.

Does 3 go into 9? Yes.

So 3 goes into our original number also.

Both 2 and 3 work, so 6 also works.
Yes.

Find all the factors of 300.
Start counting, and writing the matching factor.

1 × 300. 2 × 150. 3 × 100. 4 × 75. 5 × 60. 6 × 50. 7 doesn't work. 8 doesn't work. 9 doesn't work. 10 × 30. 11 doesn't work. 12 × 25. 13 doesn't work. 14 doesn't work. 15 × 20. 16 doesn't work. 17 doesn't work. 18 doesn't work. 19 doesn't work.

Now we are up to 20, a number that already appeared. We can stop.
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300

Find the prime factorization of 18.
Make a factor tree. However you do that, the "leaves" will be 2, 3, and 3.

So you can write 2 × 3 × 3 or you can write 2 × 3^{2}
You can write 2 × 3 × 3, or you can write 2 × 3^{2}

Find the prime factorization of 60.
Make a factor tree. However you do that, the "leaves" will be 2, 2, 3, and 5.

So you can write 2 × 2 × 3 × 5 or you can write 2^{2} × 3 × 5
You can write 2 × 2 × 3 × 5, or you can write 2^{2} × 3 × 5

(we need some problems here)

(we need some problems here)

Write the ratio "85 to 97" in fraction notation.
Simply write one number above the other, the fraction bar represents the word "to"^{85}⁄_{97}
^{85}⁄_{97}

Write the ratio "0.34 to 124" in fraction notation.
Simply write one number above the other, the fraction bar represents the word "to"^{0.34}⁄_{124}
^{0.34}⁄_{124}

A twelve pound shankless ham contains sixteen servings. What is the rate in servings per pound?
The word "per" means division. So we do servings ÷ pounds = 16 ÷ 12 ≈ **1.3 servings per pound**
1.3 servings per pound

A car will travel 464 miles on 14.5 gallons of gasoline. What is the rate in miles per gallon?
The word "per" means division. So we do miles ÷ gallon = 464 ÷ 14.5 = **32 miles per gallon**
32 miles per gallon

A sixteen ounce bag of salad greens costs $2.39. Find the unit price in cents per ounce.
The word "per" means division. So we do cents ÷ ounce = 239 ÷ 16 ≈ **15 cents per ounce**
15 cents per ounce

(we need some problems here)

Write 14.7% as a decimal.
Use RIP LOP to remind us to go "left out of percent" two scoots, to get **0.147**
0.147

Write 0.38 as a percent.
Use RIP LOP to remind us to go "right into percent" two scoots, to get **38%**
38%

Write 65% as a fraction.
Replace the % symbol with "write the number over 100", to get ^{65}⁄_{100} = ^{13}⁄_{20}^{13}⁄_{20}

Write ^{11}⁄_{8} as a percent.
First do "top ÷ bottom" to change ^{11}⁄_{8} into 1.375

Then use RIP LOP to remind us to go "right into percent" two scoots, to get **137.5%**
137.5%

(we need some problems here)

(we need some problems here)

(we need some problems here)

How many meters is 6 kilometers?
On the list of SI prefixes, move 3 scoots right from "kilo" to "plain units", to get **6,000 m**
6,000 m

How many centimeters is 8.7 millimeters?
On the list of SI prefixes, move 1 scoot left from "milli" to "centi", to get **0.87 cm**
0.87 cm

Something is 0.5 cm wide. Express this width in meters and millimeters.
For meters, on the list of SI prefixes, move 2 scoots left from "centi" to "plain units", to get **0.005 m**

For millimeters, on the list of SI prefixes, move 1 scoot right from "centi" to "milli", to get **5 mm**
0.005 m and 5 mm

Marvin L. Bittinger is 1.8542 meters tall. Express this height in centimeters and millimeters.
For centimeters, on the list of SI prefixes, move 2 scoots right from "plain units" to "centi", to get **185.42 cm**

For millimeters, on the list of SI prefixes, move 3 scoots right from "plain units" to "milli", to get **1,854.2 mm**
185.42 cm and 1,854.2 mm

How many liters is 3,080 milliliters?
On the list of SI prefixes, move 3 scoots left from "milli" to "plain units", to get **3.080 L**
3.080 L

How many milliliters is 0.25 liters?
On the list of SI prefixes, move 3 scoots right from "plain units" to "milli", to get **250 mL**
250 mL

How many grams is 3.8 kilograms?
On the list of SI prefixes, move 3 scoots right from "kilo" to "plain units", to get **3,800 g**
3,800 g

How many grams is 2,200 milligrams?
On the list of SI prefixes, move 3 scoots left from "milli" to "plain units", to get **2.200 g**
2.200 g

How many micrograms is 0.37 milligrams?
On the list of SI prefixes, move 3 scoots right from "milli" to "micro", to get **370 mcg**
370 mcg

Write as multiplication, then multiply: 3^{2} =
3^{2} = 3 × 3 = **9**
9

Twelve square feet is how many square inches?
One square foot is 12 inches × 12 inches = 144 square inches.

So twelve square feet is twelve of that: **1,728 sq. in.**
1,728 sq. in.

Three square centimeters is how many square meters?
One square meter is 100 cm × 100 cm = 10,000 square cm.

For three square cm we do 3 ÷ 10,000 = **0.0003 sq. m**
0.0003 sq. m

What is the square root of 225?
A square with sides 15 would have area 15 × 15 = 225.

So the answer is **15**
15

What is the square root of 87? (Round to the thousandths place.)
Let our calculator do the work, to find the answer of **9.327**
9.327

Solve: 6 × *n* = 24
The opposite of ×6 is ÷6. So do that to both sides and get ** n = 4**
4

Solve: 4 × *n* = 36
The opposite of ×4 is ÷4. So do that to both sides and get ** n = 9**
9

Solve: 15 = 3 × *n*
The opposite of ×3 is ÷3. So do that to both sides and get **5 = n**
5

Solve: 21 = 3 × *n*
The opposite of ×3 is ÷3. So do that to both sides and get **7 = n**
7

Thank you to the wonderful website MathTV for providing so many math problems with step-by-step video solutions!

Even better than having only videos is *also* having the textbook *Prealgebra* by the same publisher. Purchasing the textbook in either physical or e-book format provides access to even more videos. Every single example problem in that book has videos that go with it!

The 11th Edition of *Basic Mathematics* by Marvin L. Bittinger put all of its chapter test problems on YouTube. Another great source of step-by-step video solutions!

(Include other online video sources too, eventually.)