Elementary School Equation: Commented and Solved Exercises

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At first degree equations are math sentences like ax + b = 0, where a and b are real numbers and x is the unknown (unknown term).

Several types of problems are solved through this calculation, so knowing how to solve a 1st degree equation is fundamental.

Take advantage of the commented and solved exercises to exercise this important math tool.

question 1

(CEFET/RJ - 2nd phase - 2016) Carlos and Manoela are twin brothers. Half of Carlos' age plus a third of Manoela's age is equal to 10 years. What is the sum of the ages of the two brothers?

Correct answer: 24 years.

As Carlos and Manoela are twins, their ages are the same. Let's call this age x and solve the following equation:

x over 2 plus x over 3 equal to 10 numerator 3 x plus 2 x over denominator 6 end of fraction equal to 10 5 x equal to 10.6 x equal to 60 over 5 x equal to 12

Therefore, the sum of the ages is equal to 12 + 12 = 24 years.

question 2

(FAETEC - 2015) A package of the Tasty biscuit costs R$ 1.25. If João bought N packages of this cookie spending R$ 13.75, the value of N is equal to:

a) 11
b) 12
c) 13
d) 14
e) 15

Correct alternative: a) 11.

The amount spent by João is equal to the number of packages he bought times the value of 1 package, so we can write the following equation:

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1 comma 25 space. space N space equal to 13 comma 75 N equal to numerator 13 comma 75 over denominator 1 comma 25 end of fraction N equal to 11

Therefore, the value of N is equal to 11.

question 3

(IFSC - 2018) Consider the equation numerator 3 x over denominator 4 end of fraction equal to 2 x plus 5, and tick the CORRECT alternative.

a) It is a function of the first degree, its solution is = −1 and its solution set is = {−1}.
b) It is a rational equation, its solution is = −4 and its solution set is = {−4}.
c) It is an equation of the first degree, its solution is = +4 and its solution set is = ∅.
d) It is a second degree equation, its solution is = −4 and its solution set is = {−4}.
e) It is an equation of the first degree, its solution is = −4 and its solution set is = {−4}.

Correct alternative: e) It is an equation of the first degree, its solution is = −4 and its solution set is = {−4}.

The indicated equation is an equation of the first degree. Let's solve the indicated equation:

numerator 3 x over denominator 4 end of fraction equal to 2 x plus 5 2 x minus numerator 3 x over denominator 4 end of fraction equal to minus 5 numerator 8 x minus 3 x over denominator 4 end of fraction equal to minus 5 5 x equal to minus 5.4 x equal to numerator minus 20 over denominator 5 end of fraction equal to minus 4

Therefore, numerator 3 straight x over denominator 4 end of fraction equal to 2 straight x plus 5 is an equation of the first degree, its solution is = −4 and its solution set is = {−4}.

question 4

(Colégio Naval - 2016) In the exact division of the number k by 50, a person, absently, divided by 5, forgetting the zero and, thus, found a value 22.5 units higher than expected. What is the value of the tens digit of the number k?

to 1
b) 2
c) 3
d) 4
e) 5

Correct alternative: b) 2.

By writing the problem information in the form of an equation, we have:

k over 5 equals k over 50 plus 22 comma 5 k over 5 minus k over 50 equals 22 comma 5 numerator 10 k minus k over denominator 50 end of fraction equal to 22 comma 5 9 k equal to 22 comma 5.50 k equal to 1125 over 9 equal to 125

Therefore, the value of the tens digit of the number k is 2.

question 5

(Colégio Pedro II - 2015) Rosinha paid R$67.20 for a blouse that was being sold at a 16% discount. When her friends found out, they rushed to the store and had the sad news that the discount was over. The price found by Rosinha's friends was

a) BRL 70.00.
b) BRL 75.00.
c) BRL 80.00.
d) BRL 85.00.

Correct alternative: c) R$ 80.00.

Calling x the amount paid by Rosinha's friends, we can write the following equation:

x minus 16 over 100 x equal to 67 comma 2 numerator 100 x minus 16 x over denominator 100 end of fraction equal to 67 comma 2 84 x equal to 67 comma 2100 84 x equal to 6720 x equal to 6720 over 84 x equal to 80

Therefore, the price found by Rosinha's friends was R$ 80.00.

question 6

(IFS - 2015) A Teacher spends 1 third of your salary with food, 1 half with housing and still have R$ 1,200.00. What is this teacher's salary?

a) BRL 2,200.00
b) BRL 7,200.00
c) BRL 7,000.00
d) BRL 6,200.00
e) BRL 5,400.00

Correct alternative: b) BRL 7,200.00

Let's call the teacher's salary value x and solve the following equation:

1 third x plus 1 half x plus 1200 equals x x minus numerator start style show 1 end style over denominator start style show 3 end style end fraction x minus numerator start style show 1 end style over denominator start style show 2 end style end of fraction x equal to 1200 numerator 6 x minus 2 x minus 3 x over denominator 6 end of fraction equal to 1200 x over 6 equal to 1200 x equal to 7200

Therefore, this teacher's salary is R$7,200.00.

question 7

(Apprentice Sailor - 2018) Analyze the following figure.

Sailor's Apprentice Question 2018 Equation of 1st Grade

An architect intends to fix on a panel 40 m long horizontally seven engravings 4 m long horizontally each. The distance between two consecutive engravings is d, while the distance from the first and last engraving to the respective sides of the panel is 2d. Therefore, it is correct to say that d it's the same as:

a) 0.85 m
b) 1.15 m
c) 1.20 m
d) 1.25 m
e) 1.35 m

Correct alternative: c) 1.20 m.

The total length of the panel is equal to 40 m and there are 7 engravings with 4 m, so, to find the measure that will be left, we will do:

40 - 7. 4 = 40 - 28 = 12 m

Looking at the figure, we see that we have 6 spaces with a distance equal to d and 2 spaces with a distance equal to 2d. Thus, the sum of these distances must equal 12 m, so:

6d + 2. 2d = 12
6d + 4d = 12
10d = 12
d equals 12 over 10 equals 1 comma 20 space m

Therefore, it is correct to say that d is equal to 1.20 m.

question 8

(CEFET/MG - 2018) In a family with 7 children, I am the youngest and 14 years younger than my mother's eldest. Among the children, the fourth is a third of the older brother's age, plus 7 years. If the sum of our three ages is 42, then my age is a number.

a) divisible by 5.
b) divisible by 3.
c) cousin.
d) par.

Correct alternative: c) cousin.

Calling the oldest child's age x, we have the following situation:

  • eldest child: x
  • Youngest child: x - 14
  • Fourth child: x over 3 plus 7

Considering that the sum of the age of the three siblings is equal to 42, we can write the following equation:

x plus left parenthesis x minus 14 right parenthesis plus left parenthesis x over 3 plus 7 right parenthesis equals 42 2 x plus x over 3 equal to 42 minus 7 plus 14 numerator 6 x plus x over denominator 3 end of fraction equal to 49 7 x equal to 49.3 x equal to 147 over 7 x equal to 21

To find the age of the youngest, just do:

21 - 14 = 7 (prime number)

So if the sum of our three ages is 42, then my age is a prime number.

question 9

(EPCAR - 2018) A used car dealership presents a model and advertises it for x reais. To attract customers, the reseller offers two forms of payment:

Epcar Question 2018 Equation of the 1st degree

A customer purchased a car and opted to pay by credit card in 10 equal installments of R$ 3,240.00 Considering the above information, it is correct to state that

a) the value x advertised by the reseller is less than R$25,000.00.
b) if this customer had opted for cash payment, then he would have spent more than R$24,500.00 with this purchase.
c) the option that this buyer made using the credit card represented a 30% increase over the amount that would be paid in cash.
d) if the customer had paid in cash, instead of using the credit card, then he would have saved more than R$8000.00.

Correct alternative: d) if the customer had paid in cash, instead of using the credit card, then he would have saved more than R$8000.00.

Solution 1

Let's start by calculating the x value of the car. We know that the customer paid in 10 installments equal to R$3240 and that in this plan, the value of the car is increased by 20%, so:

x equal to 3240.10 minus 20 over 100 x x plus 1 fifth x equal to 32400 numerator 5 x plus x over denominator 5 end of fraction equal to 32400 6 x equal to 32400.5 x equal to 162000 over 6 x equal to 27000

Now that we know the value of the car, let's calculate how much the customer would pay if they opted for the cash plan:

27000 minus 10 over 100 27000 equal to 27000 minus 2700 space equal to 24 space 300

In this way, if the customer had paid in cash, he would have saved:

32400 - 24 300 = 8 100

Solution 2

An alternative way to solve this problem would be:

1st step: determine the amount paid.

10 installments of R$ 3 240 = 10 x 3 240 = R$ 32 400

2nd step: determine the car's original value using the rule of three.

table row with cell with 32 space 400 end of cell minus cell with 120 percent sign end of cell row with straight x minus cell with 100 percent sign end of cell row with blank blank blank row with straight x equal to cell with numerator 32 space 400 space. space 100 over denominator 120 end of fraction end of cell row with straight x equals cell with 27 space 000 end of cell end of table

Thus, as the amount paid was increased by 20%, the original price of the car is R$ 27 000.

3rd step: determine the value of the car when making the cash payment.

27 000 - 0.1 x 27 000 = 27 000 - 2 700 = 24 300

Therefore, paying cash with a 10% discount, the final value of the car would be R$ 24,300.

Step 4: Determine the difference between cash and credit card payment terms.

R$ 32 400 - R$ 24 300 = R$ 8 100

In this way, by opting for a cash purchase, the customer would have saved more than eight thousand reais in relation to the payment in installments on the credit card.

See too: Equation Systems

question 10

(IFRS-2017) Pedro had x reais from his savings. He spent a third at the amusement park with friends. The other day, he spent 10 reais on stickers for his football players album. Then he went out to have a snack with his classmates at school, spending 4/5 more than he still had and still got a change of 12 reais. What is the value of x in reais?

a) 75
b) 80
c) 90
d) 100
e) 105

Correct alternative: e) 105.

Initially, Pedro spent 1 third of x, then spent 10 reais. In the snack he spent 4 over 5 of what is left after having made the previous expenses, that is, 4 over 5 in x minus 1 third x minus 10, leaving 12 reais.

Considering this information, we can write the following equation:

1 third x plus 10 plus 4 over 5 left parenthesis x minus 1 third x minus 10 right parenthesis plus 12 space equal to x x minus 1 third x minus 4 over 5 x plus 4 over 15 x equal to 10 minus numerator 4.10 over denominator 5 end of fraction plus 12 numerator 15 x minus 5 x minus 12 x plus 4 x over denominator 15 end of fraction equal to 14 2 x equal to 210 x equal to 210 over 2 equal to 105

Therefore, the value of x in reais is 105.

Keep testing your knowledge:

  • Exercises on 1st Degree Equation with an Unknown
  • Exercises on High School Equations
  • Exercises on 1st Grade Function
  • Exercises on Rule of Three
  • Exercises on 1st Degree Equation Systems
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