Estado analítico del presupuesto de egresos, conteniendo nota explicativa

SUMMARY

KEYWORDS CAN BE misleading—sometimes even a word like

added can be found in a word problem that requires subtraction. By

carefully reading each question and understanding the context of the problem, we can choose the right operation to solve the problem. In the next chapter, we’ll look at word problems that take more than one step to solve.

ANSWERS

1. Read the entire word problem.

We are given the amount of money Rome paid for his video camera and we are given the amount less that Darryl paid than Rome.

Identify the question being asked.

We are looking for how much Darryl paid for his camera.

Underline the keywords.

The keyword phrase less than signals subtraction.

Cross out extra information and translate words into numbers.

We are told that the cameras were on sale for 15% off, but we don’t need this number to solve the problem—cross it out.

List the possible operations.

To find how much Darryl paid for his camera, we must use subtraction.

Write number sentences for each operation.

The phrase less than is a backward phrase in this problem. Although the number $45 appears first in the problem, it is $45 that must be sub- tracted from $340: $340 – $45.

Solve the number sentences and decide which answer is reasonable.

$340 – $45 = $295. Since Darryl paid less than Rome, this answer seems reasonable.

Check your work.

Since we used subtraction to find the answer to this problem, we must use addition to check our work. The amount Darryl paid, $295, plus the amount less he paid than Rome, $45, should equal the amount Rome paid, $340: $295 + $45 = $340.

2. Read the entire word problem.

We are given the numbers 14 and 168.

Identify the question being asked.

How many times 14 can divide 168?

Underline the keywords.

The keyword divide tells us to use division.

Cross out extra information and translate words into numbers.

There is no extra information and no words that need to be translated into numbers.

To find how many times 14 can divide 168, we must use division.

Write number sentences for each operation.

We could write 14  168 or 168  14, but the word divide alone with- out the word by tells us that we must reverse the order of the numbers given to us. We must find 168  14.

Solve the number sentences and decide which answer is reasonable.

168  14 = 12

Check your work.

Since we used division to find the answer to this problem, we must use multiplication to check our work. Multiply 12 by 14: 14 ≈ 12 = 168. 3. Read the entire word problem.

We are given the number of pounds of hay that are added to the stack and the original weight of the stack.

Identify the question being asked.

We are looking for the new weight of the stack.

Underline the keywords.

There are no keywords in this problem.

Cross out extra information and translate words into numbers.

The diameter of the haystack is not needed to solve this problem—cross it out.

List the possible operations.

Since 13 pounds of hay was placed on the haystack, that means that the size of the haystack is increasing. We must either add or multiply to increase the original weight of the haystack.

Write number sentences for each operation.

Addition and multiplication are commutative, so we have only one number sentence for each operation. Since 13 pounds are being added (or multiplied) to 27, the number 27 should come first in our number sentences, even though both operations are commutative: 27 + 13, 27 ≈ 13.

Solve the number sentences and decide which answer is reasonable.

27 + 13 = 40 27 ≈ 13 = 351

An answer of 351 does not seem reasonable—that would be the total weight of 13 haystacks that each weigh 27 pounds. Addition is the right operation; 40 pounds is a reasonable answer.

Check your work.

Since we used addition to find the answer to this problem, we must use subtraction to check our work. Subtract the weight added to the stack, 13 pounds, from the new weight of the stack, 40 pounds, and that should give us the original weight of the stack, 27 pounds: 40 – 13 = 27 pounds. 4. Read the entire word problem.

We are given the size of the class and the number of rows in the class.

Identify the question being asked.

We are looking for the number of students in each row.

Underline the keywords.

The keyword each signals multiplication or division.

Cross out extra information and translate words into numbers.

There is no extra information and no words that need to be translated into numbers.

List the possible operations.

We are given the total number of students, and we are looking for the number of students in each (one) row. When we are given the value of one and are looking for the total, we multiply; when we are given the total and we are looking for the value of one, we divide.

Write number sentences for each operation.

We could form the number sentence 5  30, but think about what the problem is telling us. Mr. Miller uses five rows to divide his class. When the word divide appears without the word by, we need to reverse the order of the numbers. The number sentence 30  5 should be used to solve the problem.

Solve the number sentences and decide which answer is reasonable.

30  5 = 6. It is reasonable that there are six students in each of the five rows of the classroom, since there are a total of 30 students.

Check your work.

Since we used division to find the answer to this problem, we must use multiplication to check our work. Multiply the number of rows, 5, by the number of students in each row, 6, to find the total number of stu- dents in the class, 30: 5 ≈ 6 = 30 students.

5. Read the entire word problem.

We are given the number of gumballs in a machine and the number of gumballs taken from the machine.

Identify the question being asked.

We are looking for the number of gumballs left in the machine.

Underline the keywords.

The keyword left signals subtraction.

Cross out extra information and translate words into numbers.

There is no extra information and no words that need to be translated into numbers.

List the possible operations.

Since gumballs have been taken from the machine, we must use an oper- ation that decreases the total. The keyword has already told us to use subtraction.

Write number sentences for each operation.

We could form the number sentences 13 – 76 or 76 – 13. The problem tells us that Ryan takes 13 from 76, or subtracts 13 from 76. This back- ward phrase tells us that the number sentence to use is 76 – 13.

Solve the number sentences and decide which answer is reasonable.

76 – 13 = 63. The results of 13 – 76 would be a negative number, which would not make sense. Given that the machine contained 76 gumballs before Ryan took 13, 63 is a reasonable answer.

Check your work.

Since we used addition to find the answer to this problem, we must use subtraction to check our work. Add the number of gumballs in the machine now, 63, to the number of gumballs Ryan took, 13, and that should give us the original number of gumballs in the machine, 76: 63 + 13 = 76 gumballs.

6. Read the entire word problem.

We are given the amount Bria paid for her bicycle and how much less she paid than Gavin paid for his bicycle.

Identify the question being asked.

We are looking for how much Gavin paid for his bicycle.

Underline the keywords.

The keyword phrase less than often signals subtraction.

Cross out extra information and translate words into numbers.

There is no extra information and no words that need to be translated into numbers.

Although the phrase less than could mean subtraction, we are given how much Bria paid for her bicycle and the difference between how much she paid and how much Gavin paid. Since Bria paid less, we must add how much she paid to the difference between what they paid.

Write number sentences for each operation.

$129 + $46

Solve the number sentences and decide which answer is reasonable.

$129 + $46 = $175. Since we know that Bria paid less than Gavin, it makes sense that our answer, $175, is greater than $129.

Check your work.

Since we used addition to find the answer to this problem, we must use subtraction to check our work. Subtract how much Bria paid for her bicycle, $129, from how much Gavin paid for his bicycle, $175. The dif- ference should be $46: $175 – $129 = $46.

7. Read the entire word problem.

We are given Rori’s age and how much older she is than Bret.

Identify the question being asked.

We are looking for Bret’s age.

Underline the keywords.

The keyword phrase older than often signals subtraction.

Cross out extra information and translate words into numbers.

There is no extra information and no words that need to be translated into numbers.

List the possible operations.

Since Rori is older than Bret and we are given Rori’s age, we must sub- tract the difference between their ages from Rori’s age to find Bret’s age.

Write number sentences for each operation.

The only possible number sentence is 25 – 3, since 3 – 25 would result in a negative number.

Solve the number sentences and decide which answer is reasonable.

25 – 3 = 22. Rori is older than Bret, so we expected to find an answer that is slightly less than 25.

Check your work.

Since we used subtraction to find the answer to this problem, we must use addition to check our work. Add Bret’s age to the difference between his age and Rori’s age: 22 + 3 = 25, which is Rori’s age.

8. Read the entire word problem.

We are given the rate at which an old copy machine produces copies, and we are given how much faster the new copy machine is than the old copy machine.

Identify the question being asked.

We are looking for the rate at which the new copy machine produces copies.

Underline the keywords.

The keyword phrase more than often signals subtraction.

Cross out extra information and translate words into numbers.

There is no extra information and no words that need to be translated into numbers.

List the possible operations.

The keyword phrase signals subtraction, but we are given the rate of the old machine, not the rate of the new machine. Since the new machine is faster than the old one, we must add the speed of the old machine to the difference between the speeds of the two machines.

Write number sentences for each operation.

30 + 28

Solve the number sentences and decide which answer is reasonable.

30 + 28 = 58. We expected the new machine to produce more copies per minute than the old machine, so this answer is reasonable.

Check your work.

Since we used addition to find the answer to this problem, we must use subtraction to check our work. Subtract the speed of the new machine from the speed of the old machine. The difference should be 30 copies per minute: 58 – 28 = 30 copies per minute.

9. Read the entire word problem.

We are given the year in which Arkansas became a state and how many years fewer Arizona has been a state.

Identify the question being asked.

We are looking for the year in which Arizona became a state.

Underline the keywords.

The keyword fewer often signals subtraction.

There is no extra information and no words that need to be translated into numbers.

List the possible operations.

The keyword fewer signals subtraction, but Arizona became a state later than Arkansas, which means that the year in which it became a state is a larger number than the year in which Arkansas became a state. We must add 76 to 1836.

Write number sentences for each operation.

76 + 1836

Solve the number sentences and decide which answer is reasonable.

76 + 1836 = 1912. Arizona became a state in 1912.

Check your work.

Since we used addition to find the answer to this problem, we must use subtraction to check our work. Subtract the year Arkansas became a state from the year Arizona became a state. That difference should be 76 years: 1912 – 1836 = 76 years.

10. Read the entire word problem.

We are given the distance from San Francisco to Los Angeles and how much farther that distance is than the distance from Los Angeles to San Diego.

Identify the question being asked.

We are looking for the distance from Los Angeles to San Diego.

Underline the keywords.

The keyword phrase more than often signals subtraction.

Cross out extra information and translate words into numbers.

There is no extra information and no words that need to be translated into numbers.

List the possible operations.

The keyword phrase more than signals subtraction, and since we are given the distance from San Francisco to Los Angeles, we can subtract the difference between that distance and the distance from Los Angeles to San Diego to find the distance from Los Angeles to San Diego.

Write number sentences for each operation.

389 – 265

Solve the number sentences and decide which answer is reasonable.

Check your work.

Since we used subtraction to find the answer to this problem, we must use addition to check our work. Add the distance from San Diego to Los Angeles to the difference between that distance and the distance from Los Angeles to San Francisco. The sum should be equal to the distance from Los Angeles to San Francisco: 124 + 265 = 389 miles.

11. Read the entire word problem.

We are given a product and one factor.

Identify the question being asked.

We are looking for the other factor.

Underline the keywords.

The keyword product often signals multiplication.

Cross out extra information and translate words into numbers.

There is no extra information and no words that need to be translated into numbers.

List the possible operations.

The keyword product signals multiplication, but since we are given a product, the multiplication has already taken place. In order to find a factor given the other factor and a product, we must divide the product by the given factor.

Write number sentences for each operation.

1₁7₆6

Solve the number sentences and decide which answer is reasonable.

1₁7₆6 = 11. We can decide if our answer is reasonable by checking our work.

Check your work.

Our answer is correct if 11 multiplied by 16 is 176: 11 ≈ 16 = 176, which answers the question.

12. Read the entire word problem.

We are given the number of groups of students and the number of stu- dents in each group.

Identify the question being asked.

We are looking for the total number of students.

Underline the keywords.

The keyword divided usually signals division.

The number of eighth-grade classes, 6, is not needed to solve this prob- lem, so cross out that number.

List the possible operations.

The keyword divided signals division, but the division has already occurred. The total number of students has been divided into ten groups of 18, so in order to find the total number of students, we must undo that operation and multiply 10 by 18.

Write number sentences for each operation.

10 ≈ 18

Solve the number sentences and decide which answer is reasonable. 10 ≈ 18 = 180 students. Our answer seems reasonable since we would expect the total number of students to be much larger than the num- ber of groups and the number of students in each group.

Check your work.

Since we used multiplication to find the answer to this problem, we must use division to check our work. Divide the number of students, 180, by the number of students in each group, 18. The result should be the number of groups, 10: 1₁8₈0= 10 groups.

13. Read the entire word problem.

We are given the current temperature on the lake and the change in tem- perature that led to the current temperature.

Identify the question being asked.

We are looking for the temperature prior to the snowstorm.

Underline the keywords.

The keyword dropped often signals subtraction.

Cross out extra information and translate words into numbers.

There is no extra information and no words that need to be translated into numbers.

List the possible operations.

The keyword dropped signals subtraction, but we are given the tem- perature after this subtraction, or drop, had occurred. To find the tem- perature before the temperature decreased, we must add the change, 15º, to the current temperature.

Write number sentences for each operation.

–9 + 15

–9 + 15 = 6º. Since the snowstorm dropped the temperature on the lake, we expected the original temperature to be higher than –9º, so our answer makes sense.

Check your work.

Since we used addition to find the answer to this problem, we must use subtraction to check our work. Subtract 15 from the original tempera- ture, 6º, and that should give us the current temperature, –9º: 6 – 15 = –9º.

14. Read the entire word problem.

We are given the total number of miles Jordan and Jared ran and the number of miles Jordan ran.

Identify the question being asked.

We are looking for the number of miles Jared ran.

Underline the keywords.

The keyword altogether often signals addition.

Cross out extra information and translate words into numbers.

The number of hours Jordan and Jared ran is not needed to solve the problem, so cross that detail out.

List the possible operations.

The keyword signals addition, but we already have the total number of miles they ran. To find how many miles Jared ran, we must subtract the number of miles Jordan ran from the total number of miles that they both ran.

Write number sentences for each operation.

25 – 13

Solve the number sentences and decide which answer is reasonable.

25 – 13 = 12 miles. We expected our answer to be less than 25 miles, so this answer seems reasonable.

Check your work.

Since we used subtraction to find the answer to this problem, we must use addition to check our work. Add the number of miles Jordan ran, 13, to the number of miles Jared ran, 12. The total should be 25 miles: 13 + 12 = 25 miles.

15. Read the entire word problem.

We are given the amount of money Brooke has in her purse now and the amount of money she spent.

Identify the question being asked.

We are looking for how much money she had before she went shopping.

Underline the keywords.

The keyword left usually signals subtraction, as does the word spent.

Cross out extra information and translate words into numbers.

There is no extra information and no words that need to be translated into numbers.

List the possible operations.

The keyword left signals subtraction, but we are given the amount in Brooke’s purse after she spent $78. To find how much money she had in her purse before she went shopping, we need to add $78 to the total

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