ALGEBRAIC STATEMENTS
- A statement that is Algebraic in form. It is said that the "Algebraic Statement" should be translated to "Algebraic expression or equation" for a purpose-that is to use in mathematical manipulation. This is a must when we are dealing with Algebraic worded problem. Hard. Difficult. We are not interested. But since it is a part of almost all curricula, we have to. To start tackling those Algebraic worded problem, we have to reconsider the following facts.
Facts to remember before solving Algebraic worded problems:
1. Read the problem, carefully. Do not skim! Draw a picture if you honestly don't understand what does the problem says. Check what is all about.Don't get into rush.
2. Usually, at the last part of the statement of the problem, there is/are question(s). In this case, you will have an idea, what is being "asked". You can recall what to use in your "stocked" knowledge.
3. Assign variable when you start solving the problem. Usually "Let x". If you are ask for the "age"
in the "age problem", do it like, "Let x = Amy's age"(example only), for the unknown. This is how we "assume for the meantime" what is we don't know, but need to be found--it is being "asked" by the problem. You can use other variable like "y","z", etc. there's no problem with that.
4.If the unknown is more than one, start from the smallest one.
5. Draw what is the scenario of the problem looks like. It is easy to visualize the problem aided with some illustrations. Check if your drawings corresponds to the problem scenario.
6. Read again the problem to double check. Recall your knowledge of how to attack with it.
7. If the problem has "units" like miles,oz,meter,kg,etc., do not forget to include it for your final answer as majority always do.
8. Be systematic. Suggestions will be like the following:
Given:(put the possible given found in the statement of the problem)
Required (What is asked): write what is need to be found
Solution: Show solution in logical sequence
Provided here are some illustrations of Algebraic statements translated to Algebraic expressions.equations. The red colored one is the translation from statement to expression.
Algebraic Statement Algebraic Equations
1. Twice of the unknown means Let x = unknown, 2x
2. Three less than the unknown Let y = unknown, y-3
3. A numbered decreased by 5 Let z = unknown, z-5
4. 5 decreased by the unknown Let y = unknown, 5-x
5. Six more than the unknown Let x = unknown, x + 6
6. Three more than twice the unknown Let x = unknown
2x = twice the unknown
2x + 3 = two more than twice the unknown
8. John's age 10 years ago Let x = John's age
9. Number of cents in x quarters Let x = quarters, 25x
10. Number of cents in 3x dimes dimes is 10, so 10(3x)
11. Number of cents in 3 nickels Let x = nickels, 3x
12. Number of cents in x + 5 nickels Let x = nickels, so 5(x+5)
13. Separate 10 into two parts Let x = first part, then 10 - x the second part
Therefore, x and 10 - x.
14. The distance traveled at x hours in 65 mph. Let x = speed, then 65x
15. There are two consecutive integers. Let x = the first integer, so x + 1 the second
Therefore, x and x + 1
16. There are two consecutive even integers Let x = the first integer, so x + 2 the second
Therefore, x and x + 2
17. There are odd consecutive even integers Let x = the first integer, so x + 2 the second
Therefore, x and x + 2
18. The interest on a dollar 12% for a year Let x = interest, so 0.12x
19. A money worth $1000 separated for investment Let x = first amount, then $1000-x the second
Illustrative Examples:
1. There are two numbers whose sum is 12. The other number is twice the other number. Can you find these two numbers? (Remember the facts before solving)
Given: 12 and unknown other number
Required: find these two numbers
Solution:
Let x = the first number, the smaller
2x = twice the other number, the bigger
x + 2x = 12, final equation
So, x + 2x = 12
3x = 12
x = 4
therefore the first number is 4 and the other is 2x = 2(4) =8!
let's substitute:
x + 2x = 12
4 + 2(4) = 12
4 + 8 = 12
12 = 12, if the result is equal upon checking, you're right!
2. Sixty is the sum when two numbers are added. Three times of the first number is 10 more than twice the second. What are the numbers? (Remember the facts before solving)
Given: 60 and unknown other number
Required: find these two numbers
Solution:
Let x = the first number
60 - x = the second number
2( 60 - x) = twice the second number
2(60 - x) +10 = 10 more than twice the second
3x = 2(60 - x) +10, final equation
let's work on it..
3x = 2(60 - x) +10
3x = 120 - 2x + 10
3x + 2x = 120 + 10
5x = 130
x= 26
- A statement that is Algebraic in form. It is said that the "Algebraic Statement" should be translated to "Algebraic expression or equation" for a purpose-that is to use in mathematical manipulation. This is a must when we are dealing with Algebraic worded problem. Hard. Difficult. We are not interested. But since it is a part of almost all curricula, we have to. To start tackling those Algebraic worded problem, we have to reconsider the following facts.
Facts to remember before solving Algebraic worded problems:
1. Read the problem, carefully. Do not skim! Draw a picture if you honestly don't understand what does the problem says. Check what is all about.Don't get into rush.
2. Usually, at the last part of the statement of the problem, there is/are question(s). In this case, you will have an idea, what is being "asked". You can recall what to use in your "stocked" knowledge.
3. Assign variable when you start solving the problem. Usually "Let x". If you are ask for the "age"
in the "age problem", do it like, "Let x = Amy's age"(example only), for the unknown. This is how we "assume for the meantime" what is we don't know, but need to be found--it is being "asked" by the problem. You can use other variable like "y","z", etc. there's no problem with that.
4.If the unknown is more than one, start from the smallest one.
5. Draw what is the scenario of the problem looks like. It is easy to visualize the problem aided with some illustrations. Check if your drawings corresponds to the problem scenario.
6. Read again the problem to double check. Recall your knowledge of how to attack with it.
7. If the problem has "units" like miles,oz,meter,kg,etc., do not forget to include it for your final answer as majority always do.
8. Be systematic. Suggestions will be like the following:
Given:(put the possible given found in the statement of the problem)
Required (What is asked): write what is need to be found
Solution: Show solution in logical sequence
Provided here are some illustrations of Algebraic statements translated to Algebraic expressions.equations. The red colored one is the translation from statement to expression.
Algebraic Statement Algebraic Equations
1. Twice of the unknown means Let x = unknown, 2x
2. Three less than the unknown Let y = unknown, y-3
3. A numbered decreased by 5 Let z = unknown, z-5
4. 5 decreased by the unknown Let y = unknown, 5-x
5. Six more than the unknown Let x = unknown, x + 6
6. Three more than twice the unknown Let x = unknown
2x = twice the unknown
2x + 3 = two more than twice the unknown
7. John's age 10 years from now Let x = John's age
x + 10 = John's age 10 years from now
8. John's age 10 years ago Let x = John's age
x - 10 = John's age 10 years ago
9. Number of cents in x quarters Let x = quarters, 25x
10. Number of cents in 3x dimes dimes is 10, so 10(3x)
11. Number of cents in 3 nickels Let x = nickels, 3x
12. Number of cents in x + 5 nickels Let x = nickels, so 5(x+5)
13. Separate 10 into two parts Let x = first part, then 10 - x the second part
Therefore, x and 10 - x.
14. The distance traveled at x hours in 65 mph. Let x = speed, then 65x
15. There are two consecutive integers. Let x = the first integer, so x + 1 the second
Therefore, x and x + 1
16. There are two consecutive even integers Let x = the first integer, so x + 2 the second
Therefore, x and x + 2
17. There are odd consecutive even integers Let x = the first integer, so x + 2 the second
Therefore, x and x + 2
18. The interest on a dollar 12% for a year Let x = interest, so 0.12x
19. A money worth $1000 separated for investment Let x = first amount, then $1000-x the second
Therefore, x and $1000-x
20. The distance traveled at 2 hours Let x = speed, then 2x
21. The distance traveled in 90 minutes at x mph Let x = speed, then 1.5x
22. The sum of a certain number and 5 Let x = the number, then x + 5
23. The product of a number and 7 Let x = the number, then 7x
24. The quotient of a number and 10 Let x = the number, then x/10
25. A number is three times as much Let x = the number, then 3x
Additional Info:
1. "is, will be, was, becomes" means "equal"
2. "More than, will be added, increased, sum, augmented" means "add"
3. "Less than, diminished, subtracted" means "subtract"
4. "Separate" means "two numbers to be added to satisfy the whole"
5. "Of" means "Multiply"
Preparatory Examples:
1. Two numbers whose sum is 15 (Remember the facts before solving)
Given: x as unknown number and 15
Required: to represent the two numbers that when added will become 15
Solution:
Let x = the unknown number
15-x = when unknown number subtracted from known number, it is the whole number
So the numbers are x and 15 - x
Try putting any values to x not greater than 15
2. A father is 5 years older than twice the age of his daughter. (Remember the facts before solving)
Given: x as unknown daughter's age and 5 years Dad's older age than twice of his daughters'
Required: to represent their ages Algebraically
Solution:
Let x = Daughter's age
2x = twice of the daughter's age
2x + 5 = Dad's age
3. A man traveling at a constant speed for 2 hours. (Remember the facts before solving)
Given: x as mph rate and 2 as number of hours traveled
Required: to represent mph rate and number of miles traveled
Solution:
Let x = rate in mph
2x = number of miles traveled
1. Two numbers whose sum is 15 (Remember the facts before solving)
Given: x as unknown number and 15
Required: to represent the two numbers that when added will become 15
Solution:
Let x = the unknown number
15-x = when unknown number subtracted from known number, it is the whole number
So the numbers are x and 15 - x
Try putting any values to x not greater than 15
2. A father is 5 years older than twice the age of his daughter. (Remember the facts before solving)
Given: x as unknown daughter's age and 5 years Dad's older age than twice of his daughters'
Required: to represent their ages Algebraically
Solution:
Let x = Daughter's age
2x = twice of the daughter's age
2x + 5 = Dad's age
3. A man traveling at a constant speed for 2 hours. (Remember the facts before solving)
Given: x as mph rate and 2 as number of hours traveled
Required: to represent mph rate and number of miles traveled
Solution:
Let x = rate in mph
2x = number of miles traveled
Illustrative Examples:
1. There are two numbers whose sum is 12. The other number is twice the other number. Can you find these two numbers? (Remember the facts before solving)
Given: 12 and unknown other number
Required: find these two numbers
Solution:
Let x = the first number, the smaller
2x = twice the other number, the bigger
x + 2x = 12, final equation
So, x + 2x = 12
3x = 12
x = 4
therefore the first number is 4 and the other is 2x = 2(4) =8!
let's substitute:
x + 2x = 12
4 + 2(4) = 12
4 + 8 = 12
12 = 12, if the result is equal upon checking, you're right!
2. Sixty is the sum when two numbers are added. Three times of the first number is 10 more than twice the second. What are the numbers? (Remember the facts before solving)
Given: 60 and unknown other number
Required: find these two numbers
Solution:
Let x = the first number
60 - x = the second number
2( 60 - x) = twice the second number
2(60 - x) +10 = 10 more than twice the second
3x = 2(60 - x) +10, final equation
let's work on it..
3x = 2(60 - x) +10
3x = 120 - 2x + 10
3x + 2x = 120 + 10
5x = 130
x= 26
Knowing that our first number is x, it is 26, and the second is 60 - x, it follows that 60 - 26 is 34.
Hence, the numbers are 26 and 34. Let's add, 26 + 34 = 60 which satisfies the statement in the problem.
Let's check!
3x = 2(60 - x) +10, final equation
3(26) = 2(60 - 26) + 10, by substitution of x value
78 = 2 (34) + 10
78 = 68 + 10
78 = 78, if the result is equal upon checking, you're right!
Hence, the numbers are 26 and 34. Let's add, 26 + 34 = 60 which satisfies the statement in the problem.
Let's check!
3x = 2(60 - x) +10, final equation
3(26) = 2(60 - 26) + 10, by substitution of x value
78 = 2 (34) + 10
78 = 68 + 10
78 = 78, if the result is equal upon checking, you're right!
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