Operations with Whole Numbers

Addition is joining groups together to find the total, or sum. The plus sign (\(+\)) tells us to add.
We can show "\(\textcolor{colordef}{\text{two hundred eighteen}}\) plus \(\text{\textcolor{colorprop}{one hundred twenty-three}}\) equals \(\text{\textcolor{olive}{three hundred forty-one}}\)" in many ways:

• With numbers: $$\textcolor{colordef}{218} + \textcolor{colorprop}{123} = \textcolor{olive}{341}$$
• With cubes:\(+\)\(=\)
• With a column addition:
• With a part-whole model:

Let's calculate: $$\textcolor{colordef}{288} + \textcolor{colorprop}{545}$$

• Step 1: Line up the numbers by place value.
  Write the numbers one on top of the other. Make sure the ones, tens, and hundreds are in straight columns.
• Step 2: Add the ones. $$\textcolor{colordef}{8} \text{ ones} + \textcolor{colorprop}{5} \text{ ones} = 13 \text{ ones}$$ 13 ones is the same as 1 ten and 3 ones. We write the \(\textcolor{olive}{3}\) in the ones place and carry the 1 ten over to the tens column.
• Step 3: Add the tens. $$1 \text{ ten (carried)} + \textcolor{colordef}{8} \text{ tens} + \textcolor{colorprop}{4} \text{ tens} = 13 \text{ tens}$$ 13 tens is the same as 1 hundred and 3 tens. We write the \textcolor{olive}{3} in the tens place and carry the 1 hundred over.
• Step 4: Add the hundreds. $$1 \text{ hundred (carried)} + \textcolor{colordef}{2} \text{ hundreds} + \textcolor{colorprop}{5} \text{ hundreds} = \textcolor{olive}{8} \text{ hundreds}$$ Write the \(\textcolor{olive}{8}\) in the hundreds place.
• The Result: $$\textcolor{colordef}{288} + \textcolor{colorprop}{545} = \textcolor{olive}{833}$$

\(+\) \(=\)

Calculate \(189+784\)

Subtraction means taking an amount away from a group to find out what is left. This result is called the difference. The minus sign (\(-\)) tells us to subtract.
We can show "two hundred thirty-seven minus one hundred fourteen equals one hundred twenty-three" in different ways:

• With Numbers: $$237 - 114 = 123$$
• With Cubes: \(-\) \(=\)
• With Column Subtraction:
• With a Part-Whole Model:

Let's calculate:\(32 - 14\)

• Step 1: Set up the subtraction.
• Step 2: Check the ones place. We need to do \(2 - 4\), but we don't have enough ones.
• Step 3: Compensate. Add 10 to both numbers (10 ones to the top; 1 ten to the bottom)
• Step 4: Subtract the ones. Now it's easy: $$12 \text{ ones} - 4 \text{ ones} = 8 \text{ ones}$$
• Step 5: Subtract the tens.$$3 \text{ tens (from 32)} - 1 \text{ ten (from 14)} - 1 \text{ ten (compensated)} = 1 \text{ ten}$$
• Result: The difference is 1 ten and 8 ones. So, $$32 - 14 = 18$$

\(-\) \(=\)

Calculate \(784 - 189\)

Multiplication is a fast way to show repeated addition. We can show the idea of "four times three equals twelve" in many different ways:

• With Numbers: $$\textcolor{colordef}{4}\times \textcolor{colorprop}{3}=\textcolor{olive}{12}$$
• In Groups: $$\textcolor{colordef}{4}\text{ groups of }\textcolor{colorprop}{3}=\textcolor{olive}{12}$$
• As Repeated Addition: $$\textcolor{colorprop}{3}+\textcolor{colorprop}{3}+\textcolor{colorprop}{3}+\textcolor{colorprop}{3}=\textcolor{olive}{12}$$
• With Cubes:
• With a Part-Whole Model:

To calculate \(23 \times 37\):

1. Step 1: Align the numbers vertically by place value.
2. Step 2: Multiply by the ones digit. Multiply the top number (\(\textcolor{colordef}{23}\)) by the ones digit of the bottom number (\(\textcolor{colordef}{7}\)): \(\textcolor{colordef}{23}\times\textcolor{colordef}{7}=\textcolor{colorprop}{161}\).
3. Step 3: Multiply by the tens digit.
   • First, place a placeholder \(\textcolor{olive}{0}\) in the ones column of the second row. This is because we are now multiplying by the tens digit (\(\textcolor{colordef}{3}\), which represents 30). This placeholder shifts our answer one place to the left.
   • Next, multiply the top number (23) by the tens digit (\(\textcolor{colordef}{3}\)). Calculate \(\textcolor{colordef}{23}\times\textcolor{colordef}{3}=\textcolor{colorprop}{69}\) and write it to the left of the placeholder.
   
4. Step 4: Sum the partial products. Add the results from Step 2 and Step 3: \(\textcolor{colordef}{161}+\textcolor{colordef}{690}=\textcolor{colorprop}{851}\)
5. Result: \(23 \times 37 = 851\).

Calculate \(123 \times 21\)

Division is the inverse operation of multiplication. It is the process of determining how many times one number is contained within another.
The components of a division expression are formally named:

• The dividend: the number that is being divided.
• The divisor: the number by which the dividend is divided.
• The quotient: the result of the division.

The operation is denoted by the division symbol (\(\div\)).$$\textcolor{olive}{\text{Dividend}} \div \textcolor{colordef}{\text{Divisor}} = \textcolor{colorprop}{\text{Quotient}}$$ For example, the multiplication fact \(\textcolor{colordef}{3} \times \textcolor{colorprop}{2} = \textcolor{olive}{6}\) corresponds to:$$\underbrace{\textcolor{olive}{6}}_{\textcolor{olive}{\text{Dividend}}} \div \underbrace{\textcolor{colordef}{3}}_{\textcolor{colordef}{\text{Divisor}}} = \underbrace{\textcolor{colorprop}{2}}_{\textcolor{colorprop}{\text{Quotient}}}.$$Division can be represented in several ways:

• Numerical Form: $$\textcolor{olive}{6}\div \textcolor{colordef}{3}=\textcolor{colorprop}{2}$$
• Word Form: Six divided by three equals two.
• Grid Model:

Euclidean Division is the process of dividing one integer (the dividend) by another (the divisor) when the division is not exact. This process yields an integer quotient and a remainder.
The components of a Euclidean division expression are formally named:

• The dividend: the number that is being divided.
• The divisor: the number by which the dividend is divided.
• The quotient: the whole number of times the divisor fits into the dividend.
• The remainder: the amount left over after the division.

This relationship is defined by the identity:$$\textcolor{olive}{\text{Dividend}} = (\textcolor{colordef}{\text{Divisor}} \times \textcolor{colorprop}{\text{Quotient}}) + \textcolor{orange}{\text{Remainder}}$$Important rules:

• The remainder is always smaller than the divisor. (If it isn’t, you can still make another group!)
• If the remainder is \(0\), the division is exact (no remainder).

Euclidean division can be represented in several ways:

• Word Form: Seven divided by three equals two, with a remainder of \(\textcolor{orange}{\text{one}}\)
• Division Sentence:$$\underbrace{\textcolor{olive}{7}}_{\textcolor{olive}{\text{Dividend}}} \div \underbrace{\textcolor{colordef}{3}}_{\textcolor{colordef}{\text{Divisor}}} = \divionRemainder{\underbrace{\textcolor{colorprop}{2}}_{\textcolor{colorprop}{\text{Quotient}}}}{\underbrace{\textcolor{orange}{1}}_{\textcolor{orange}{\text{Remainder}}}}$$
• Euclidean Identity:$$\underbrace{\textcolor{olive}{7}}_{\textcolor{olive}{\text{Dividend}}}= (\underbrace{\textcolor{colordef}{3}}_{\textcolor{colordef}{\text{Divisor}}} \times \underbrace{\textcolor{colorprop}{2}}_{\textcolor{colorprop}{\text{Quotient}}}) +\underbrace{\textcolor{orange}{1}}_{\textcolor{orange}{\text{Remainder}}}$$
• Group Model:
• Long Division Algorithm:

To divide with a remainder, like \(\textcolor{olive}{130} \div \textcolor{colordef}{4}\), follow these steps:

• Set up: Write the dividend (\(\textcolor{olive}{130}\)) inside the bracket and the divisor (\(\textcolor{colordef}{4}\)) outside.
• Divide the first part (13): “How many times does \(4\) go into \(13\)?”$$\textcolor{colordef}{4}\times \textcolor{colorprop}{3}=\boxed{12}\ (\le \textcolor{olive}{13}),\qquad\textcolor{colordef}{4}\times \textcolor{colorprop}{4}=\cancel{16}\ (> \textcolor{olive}{13}).$$Write \(\textcolor{colorprop}{3}\) above and \(12\) under \(13\); then subtract.
• Subtract and Bring down the next digit: \(\textcolor{olive}{13}-12=\textcolor{orange}{1}\); bring down \(0\) to make \(\textcolor{olive}{10}\).
• Divide the new number (10): “How many times does \(4\) go into \(10\)?”$$\textcolor{colordef}{4}\times \textcolor{colorprop}{2}=\boxed{8}\ (\le \textcolor{olive}{10}),\qquad\textcolor{colordef}{4}\times \textcolor{colorprop}{3}=\cancel{12}\ (> \textcolor{olive}{10}).$$Write \(\textcolor{colorprop}{2}\) above, put \(8\) under \(10\), and subtract to get the remainder.
• Final answer: \(\textcolor{olive}{130} \div \textcolor{colordef}{4}= \divionRemainder{\textcolor{colorprop}{32}}{\textcolor{orange}{2}}\), and\(\textcolor{olive}{130}=\textcolor{colordef}{4}\times\textcolor{colorprop}{32}+\textcolor{orange}{2}\).

Calculate \(125 \div 4\)

To evaluate an expression, we follow these steps in order:

1. Parentheses (): Always solve what's inside parentheses first.
2. Multiplication (\(\times\)) and Division (\(\div\)): Do them next, working from left to right.
3. Addition (+) and Subtraction (-): Do them last, also working from left to right.

Calculate \(4+2 \times 3\)

To solve a word problem systematically, the following five-step procedure can be applied:

1. Understand the Goal: Read the problem carefully to identify the main question and the information provided.
2. Plan the Steps: Determine the sequence of mathematical operations required to reach the solution.
3. Write the Expression: Translate the planned steps into a single mathematical expression, using parentheses where necessary to ensure the correct order of operations.
4. Calculate the Solution: Evaluate the expression to find the numerical answer.
5. State the Conclusion: Write a final sentence that directly answers the question asked in the problem.

Hugo is planning his birthday party and needs to purchase a cake and juice.The items on his list are:

1. Cakes: 2 cakes at a cost of \(\dollar\)10 each.
2. Juice: 4 cans of juice at a cost of \(\dollar\)3 each.

Hugo has a budget of \(\dollar\)30. Determine if he can afford to purchase all the items.