22. Problems involving addition

 22. Problems involving addition - step by step solution procedure and detailed explanation for each problem.

Problem 1: Adding Fractions

Problem: A recipe requires

$\frac{1}{4}$\frac{1}{4}

cup of sugar for the cake and

$\frac{3}{4}$\frac{3}{4}

cup for the frosting. How much sugar is needed in total?

Step-by-Step Solution:

1. Identify the fractions:
   • Cake:

     $\frac{1}{4}$\frac{1}{4}
   • Frosting:

     $\frac{3}{4}$\frac{3}{4}
2. Check the denominators:
   • Both have the same denominator (4).
3. Add the numerators:
   • $1 + 3 = 4$1 + 3 = 4

     .
4. Keep the denominator:
   • $\frac{4}{4} = 1$\frac{4}{4} = 1

     .
5. Write the answer: The recipe needs 1 cup of sugar.

Detailed Explanation:

• When adding fractions with the same denominator, add the numerators and keep the denominator.
• Here,

  $\frac{1}{4} + \frac{3}{4} = \frac{1+3}{4} = \frac{4}{4} = 1$\frac{1}{4} + \frac{3}{4} = \frac{1+3}{4} = \frac{4}{4} = 1

  .
• The result, 1 cup, is a whole number, indicating the total sugar required.

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Problem 2: Adding Mixed Numbers

Problem: A carpenter cuts two pieces of wood: one is

$2 \frac{1}{3}$2 \frac{1}{3}

feet and the other is

$1 \frac{2}{3}$1 \frac{2}{3}

feet. What is the total length?

Step-by-Step Solution:

1. Identify the mixed numbers:
   • First piece:

     $2 \frac{1}{3}$2 \frac{1}{3}
   • Second piece:

     $1 \frac{2}{3}$1 \frac{2}{3}
2. Add the whole numbers:
   • $2 + 1 = 3$2 + 1 = 3

     .
3. Add the fractions:
   • $\frac{1}{3} + \frac{2}{3} = \frac{1+2}{3} = \frac{3}{3} = 1$\frac{1}{3} + \frac{2}{3} = \frac{1+2}{3} = \frac{3}{3} = 1

     .
4. Add the fraction result to the whole number:
   • $3 + 1 = 4$3 + 1 = 4

     .
5. Write the answer: The total length is 4 feet.

Detailed Explanation:

• Mixed numbers are added by separately handling the whole numbers and fractions.
• The fractions have the same denominator, so add the numerators:

  $\frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1$\frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1

  .
• This 1 is a whole number, which is added to the sum of the whole numbers (3), giving 4.
• The result, 4 feet, is the total length of the wood.

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Problem 3: Word Problem with Addition

Problem: A school collects 245 cans of food on Monday, 198 cans on Tuesday, and 312 cans on Wednesday. How many cans are collected in total?

Step-by-Step Solution:

1. Identify the quantities:
   • Monday: 245 cans
   • Tuesday: 198 cans
   • Wednesday: 312 cans
2. Set up the addition: ``` 245 +198 +312
   
   

1. Add step-by-step:
   • First, add

     $245 + 198$245 + 198

     :
     • Units:

       $5 + 8 = 13$5 + 8 = 13

       , write 3, carry 1.
     • Tens:

       $4 + 9 + 1 = 14$4 + 9 + 1 = 14

       , write 4, carry 1.
     • Hundreds:

       $2 + 1 + 1 = 4$2 + 1 + 1 = 4

       .
     • Result: 443.
   • Then, add

     $443 + 312$443 + 312

     :
     • Units:

       $3 + 2 = 5$3 + 2 = 5

       .
     • Tens:

       $4 + 1 = 5$4 + 1 = 5

       .
     • Hundreds:

       $4 + 3 = 7$4 + 3 = 7

       .
     • Result: 755.
2. Write the answer: The school collects 755 cans.

Detailed Explanation:

• This problem involves adding three multi-digit numbers, which can be done by pairing numbers for simplicity.
• Adding 245 and 198 gives 443, then adding 312 to 443 gives 755.
• Vertical addition ensures accuracy by aligning place values.
• The total, 755 cans, represents the sum of all collections.

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Problem 10: Adding Negative Numbers

Problem: A submarine is at a depth of -150 meters. It ascends 75 meters and then descends 50 meters. What is its final depth?

Step-by-Step Solution:

1. Identify the initial depth and changes:
   • Initial depth: -150 meters
   • Ascent: +75 meters (positive because it reduces depth)
   • Descent: -50 meters (negative because it increases depth)
2. Set up the addition:
   • Final depth =

     $-150 + 75 + (-50)$-150 + 75 + (-50)

     .
3. Add step-by-step:
   • First,

     $-150 + 75$-150 + 75

     :
     • $-150 + 75 = -75$-150 + 75 = -75

       (since 150 - 75 = 75, and the negative sign remains).
   • Then,

     $-75 + (-50)$-75 + (-50)

     :
     • $-75 - 50 = -125$-75 - 50 = -125

       .
4. Write the answer: The final depth is -125 meters.

Detailed Explanation:

• This problem involves adding integers, including negative numbers.
• Ascending 75 meters reduces the depth, so it’s +75. Descending 50 meters increases the depth, so it’s -50.
• Adding

  $-150 + 75 = -75$-150 + 75 = -75

  , then

  $-75 - 50 = -125$-75 - 50 = -125

  .
• The final depth, -125 meters, indicates the submarine is 125 meters below sea level.