Simplify a quadratic expression online. Online calculator. Simplifying a polynomial. Multiplying polynomials

Often tasks require a simplified answer. Although both simplified and unsimplified answers are correct, your instructor may lower your grade if you do not simplify your answer. Moreover, the simplified mathematical expression is much easier to work with. Therefore, it is very important to learn to simplify expressions.

Steps

Correct order of mathematical operations

Remember the correct order for performing mathematical operations. When simplifying a mathematical expression, you need to follow a certain order of operations, since some mathematical operations take precedence over others and must be done first (in fact, not following the correct order of operations will lead you to an incorrect result). Remember the following order of mathematical operations: expression in parentheses, exponentiation, multiplication, division, addition, subtraction.
- Note that knowing the correct order of operations will allow you to simplify most simple expressions, but to simplify a polynomial (an expression with a variable) you need to know special tricks (see the next section).
Start by solving the expression in parentheses. In mathematics, parentheses indicate that the expression within them must be evaluated first. Therefore, when simplifying any mathematical expression, start by solving the expression enclosed in parentheses (it does not matter what operations you need to perform inside the parentheses). But remember that when working with an expression enclosed in brackets, you must follow the order of operations, that is, the terms in brackets are first multiplied, divided, added, subtracted, and so on.
- For example, let's simplify the expression 2x + 4(5 + 2) + 3 2 - (3 + 4/2). Here we start with the expressions in brackets: 5 + 2 = 7 and 3 + 4/2 = 3 + 2 =5.
  - The expression in the second pair of parentheses simplifies to 5 because 4/2 must be divided first (according to the correct order of operations). If you do not follow this order, you will get the wrong answer: 3 + 4 = 7 and 7 ÷ 2 = 7/2.
- If there is another pair of parentheses in the parentheses, start simplifying by solving the expression in the inner parentheses and then move on to solving the expression in the outer parentheses.
Exponentiate. Having solved the expressions in parentheses, move on to exponentiation (remember that a power has an exponent and a base). Raise the corresponding expression (or number) to a power and substitute the result into the expression given to you.
- In our example, the only expression (number) to the power is 3 2: 3 2 = 9. In the expression given to you, replace 3 2 with 9 and you will get: 2x + 4(7) + 9 - 5.
Multiply. Remember that the multiplication operation can be represented by the following symbols: "x", "∙" or "*". But if there are no symbols between the number and the variable (for example, 2x) or between the number and the number in parentheses (for example, 4(7)), then this is also a multiplication operation.
- In our example, there are two multiplication operations: 2x (two multiplied by the variable “x”) and 4(7) (four multiplied by seven). We don't know the value of x, so we'll leave the expression 2x as is. 4(7) = 4 x 7 = 28. Now you can rewrite the expression given to you as follows: 2x + 28 + 9 - 5.
Divide. Remember that the division operation can be represented by the following symbols: “/”, “÷” or “–” (you may see this last character in fractions). For example, 3/4 is three divided by four.
- In our example, there is no longer a division operation, since you already divided 4 by 2 (4/2) when solving the expression in parentheses. So you can move on to the next step. Remember that most expressions do not contain all the mathematical operations (only some of them).
Fold. When adding terms of an expression, you can start with the term on the farthest (to the left), or you can add the terms that add easily first. For example, in the expression 49 + 29 + 51 +71, it is first easier to add 49 + 51 = 100, then 29 + 71 = 100 and finally 100 + 100 = 200. It is much more difficult to add like this: 49 + 29 = 78; 78 + 51 = 129; 129 + 71 = 200.
- In our example 2x + 28 + 9 + 5 there are two addition operations. Let's start with the outermost (left) term: 2x + 28; you can't add 2x and 28 because you don't know the value of the variable "x". Therefore, add 28 + 9 = 37. Now the expression can be rewritten as follows: 2x + 37 - 5.
Subtract. This is the last operation in the correct order of performing mathematical operations. At this stage, you can also add negative numbers or do this at the stage of adding terms - this will not affect the final result in any way.
- In our example 2x + 37 - 5 there is only one subtraction operation: 37 - 5 = 32.
At this stage, after doing all the mathematical operations, you should get a simplified expression. But if the expression given to you contains one or more variables, then remember that the term with the variable will remain as it is. Solving (not simplifying) an expression with a variable involves finding the value of that variable. Sometimes variable expressions can be simplified using special methods (see next section).
- In our example, the final answer is 2x + 32. You cannot add the two terms until you know the value of the variable "x". Once you know the value of the variable, you can easily simplify this binomial.
Simplifying complex expressions
1. Addition of similar terms. Remember that you can only subtract and add similar terms, that is, terms with the same variable and the same exponent. For example, you can add 7x and 5x, but you cannot add 7x and 5x 2 (since the exponents are different).
  - This rule also applies to members with multiple variables. For example, you can add 2xy 2 and -3xy 2 , but you cannot add 2xy 2 and -3x 2 y or 2xy 2 and -3y 2 .
  - Let's look at an example: x 2 + 3x + 6 - 8x. Here the like terms are 3x and 8x, so they can be added together. A simplified expression looks like this: x 2 - 5x + 6.
2. Simplify the number fraction. In such a fraction, both the numerator and the denominator contain numbers (without a variable). A number fraction can be simplified in several ways. First, simply divide the denominator by the numerator. Second, factor the numerator and denominator and cancel the like factors (since dividing a number by itself will give you 1). In other words, if both the numerator and denominator have the same factor, you can drop it and get a simplified fraction.
  - For example, consider the fraction 36/60. Using a calculator, divide 36 by 60 to get 0.6. But you can simplify this fraction in another way by factoring the numerator and denominator: 36/60 = (6x6)/(6x10) = (6/6)*(6/10). Since 6/6 = 1, the simplified fraction is: 1 x 6/10 = 6/10. But this fraction can also be simplified: 6/10 = (2x3)/(2*5) = (2/2)*(3/5) = 3/5.
3. If a fraction contains a variable, you can cancel like factors with the variable. Factor both the numerator and denominator and cancel the like factors, even if they contain the variable (remember that the like factors here may or may not contain the variable).
  - Let's look at an example: (3x 2 + 3x)/(-3x 2 + 15x). This expression can be rewritten (factored) in the form: (x + 1)(3x)/(3x)(5 - x). Since the 3x term is in both the numerator and denominator, you can cancel it out to give a simplified expression: (x + 1)/(5 - x). Let's look at another example: (2x 2 + 4x + 6)/2 = (2(x 2 + 2x + 3))/2 = x 2 + 2x + 3.
  - Please note that you cannot cancel any terms - only identical factors that are present in both the numerator and denominator are canceled. For example, in the expression (x(x + 2))/x, the variable (factor) “x” is in both the numerator and the denominator, so “x” can be reduced to obtain a simplified expression: (x + 2)/1 = x + 2. However, in the expression (x + 2)/x, the variable “x” cannot be reduced (since “x” is not a factor in the numerator).
4. Open parenthesis. To do this, multiply the term outside the brackets by each term in the brackets. Sometimes this helps to simplify a complex expression. This applies to both members that are prime numbers and members that contain a variable.
  - For example, 3(x 2 + 8) = 3x 2 + 24, and 3x(x 2 + 8) = 3x 3 + 24x.
  - Please note that in fractional expressions there is no need to open parentheses if both the numerator and denominator have the same factor. For example, in the expression (3(x 2 + 8))/3x there is no need to expand the parentheses, since here you can cancel the factor of 3 and get the simplified expression (x 2 + 8)/x. This expression is easier to work with; if you were to expand the parentheses, you would get the following complex expression: (3x 3 + 24x)/3x.
5. Factor polynomials. Using this method, you can simplify some expressions and polynomials. Factoring is the opposite operation of opening parentheses, that is, an expression is written as the product of two expressions, each enclosed in parentheses. In some cases, factoring allows you to reduce the same expression. In special cases (usually quadratic equations), factoring will allow you to solve the equation.
  - Consider the expression x 2 - 5x + 6. It is factored: (x - 3)(x - 2). Thus, if, for example, the expression is given (x 2 - 5x + 6)/(2(x - 2)), then you can rewrite it as (x - 3)(x - 2)/(2(x - 2)), reduce the expression (x - 2) and obtain a simplified expression (x - 3)/2.
  - Factoring polynomials is used to solve (find roots) equations (an equation is a polynomial equal to 0). For example, consider the equation x 2 - 5x + 6 = 0. By factoring it, you get (x - 3)(x - 2) = 0. Since any expression multiplied by 0 is equal to 0, we can write it like this : x - 3 = 0 and x - 2 = 0. Thus, x = 3 and x = 2, that is, you have found two roots of the equation given to you.

Using any language, you can express the same information in different words and phrases. Mathematical language is no exception. But the same expression can be equivalently written in different ways. And in some situations, one of the entries is simpler. We'll talk about simplifying expressions in this lesson.

People communicate in different languages. For us, an important comparison is the pair “Russian language - mathematical language”. The same information can be communicated in different languages. But, besides this, it can be pronounced in different ways in one language.

For example: “Petya is friends with Vasya”, “Vasya is friends with Petya”, “Petya and Vasya are friends”. Said differently, but the same thing. From any of these phrases we would understand what we are talking about.

Let's look at this phrase: “The boy Petya and the boy Vasya are friends.” We understand what we are talking about. However, we don't like the sound of this phrase. Can't we simplify it, say the same thing, but simpler? “Boy and boy” - you can say once: “Boys Petya and Vasya are friends.”

“Boys”... Isn’t it clear from their names that they are not girls? We remove the “boys”: “Petya and Vasya are friends.” And the word “friends” can be replaced with “friends”: “Petya and Vasya are friends.” As a result, the first, long, ugly phrase was replaced with an equivalent statement that is easier to say and easier to understand. We have simplified this phrase. To simplify means to say it more simply, but not to lose or distort the meaning.

In mathematical language, roughly the same thing happens. The same thing can be said, written differently. What does it mean to simplify an expression? This means that for the original expression there are many equivalent expressions, that is, those that mean the same thing. And from all this variety we must choose the simplest, in our opinion, or the most suitable for our further purposes.

For example, consider the numeric expression . It will be equivalent to .

It will also be equivalent to the first two: .

It turns out that we have simplified our expressions and found the shortest equivalent expression.

For numeric expressions, you always need to perform all the steps and get the equivalent expression as a single number.

Let's look at an example of a literal expression . Obviously, it will be simpler.

When simplifying literal expressions, it is necessary to perform all possible actions.

Is it always necessary to simplify an expression? No, sometimes it will be more convenient for us to have an equivalent but longer entry.

Example: you need to subtract a number from a number.

It is possible to calculate, but if the first number were represented by its equivalent notation: , then the calculations would be instantaneous: .

That is, a simplified expression is not always beneficial for us for further calculations.

Nevertheless, very often we are faced with a task that just sounds like “simplify the expression.”

Simplify the expression: .

Solution

1) Perform the actions in the first and second brackets: .

2) Let's calculate the products: .

Obviously, the last expression has a simpler form than the initial one. We've simplified it.

In order to simplify the expression, it must be replaced with an equivalent (equal).

To determine the equivalent expression you need:

1) perform all possible actions,

2) use the properties of addition, subtraction, multiplication and division to simplify calculations.

Properties of addition and subtraction:

1. Commutative property of addition: rearranging the terms does not change the sum.

2. Combinative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third numbers to the first number.

3. The property of subtracting a sum from a number: to subtract a sum from a number, you can subtract each term separately.

Properties of multiplication and division

1. Commutative property of multiplication: rearranging the factors does not change the product.

2. Combinative property: to multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor.

3. Distributive property of multiplication: in order to multiply a number by a sum, you need to multiply it by each term separately.

Let's see how we actually do mental calculations.

Calculate:

Solution

1) Let's imagine how

2) Let's imagine the first factor as a sum of bit terms and perform the multiplication:

3) you can imagine how and perform multiplication:

4) Replace the first factor with an equivalent sum:

The distribution law can also be used in the opposite direction: .

Follow these steps:

1) 2)

Solution

1) For convenience, you can use the distributive law, only use it in the opposite direction - take the common factor out of brackets.

2) Let’s take the common factor out of brackets

It is necessary to buy linoleum for the kitchen and hallway. Kitchen area - , hallway - . There are three types of linoleums: for, and rubles for. How much will each of the three types of linoleum cost? (Fig. 1)

Rice. 1. Illustration for the problem statement

Solution

Method 1. You can separately find out how much money it will take to buy linoleum for the kitchen, and then put it in the hallway and add up the resulting products.

For example, consider the numeric expression . It will be equivalent to .

It will also be equivalent to the first two: .

It turns out that we have simplified our expressions and found the shortest equivalent expression.

For numeric expressions, you always need to perform all the steps and get the equivalent expression as a single number.

Let's look at an example of a literal expression . Obviously, it will be simpler.

When simplifying literal expressions, it is necessary to perform all possible actions.

Is it always necessary to simplify an expression? No, sometimes it will be more convenient for us to have an equivalent but longer entry.

Example: you need to subtract a number from a number.

It is possible to calculate, but if the first number were represented by its equivalent notation: , then the calculations would be instantaneous: .

That is, a simplified expression is not always beneficial for us for further calculations.

Nevertheless, very often we are faced with a task that just sounds like “simplify the expression.”

Simplify the expression: .

Solution

1) Perform the actions in the first and second brackets: .

2) Let's calculate the products: .

Obviously, the last expression has a simpler form than the initial one. We've simplified it.

In order to simplify the expression, it must be replaced with an equivalent (equal).

To determine the equivalent expression you need:

1) perform all possible actions,

2) use the properties of addition, subtraction, multiplication and division to simplify calculations.

Properties of addition and subtraction:

1. Commutative property of addition: rearranging the terms does not change the sum.

2. Combinative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third numbers to the first number.

3. The property of subtracting a sum from a number: to subtract a sum from a number, you can subtract each term separately.

Properties of multiplication and division

1. Commutative property of multiplication: rearranging the factors does not change the product.

2. Combinative property: to multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor.

3. Distributive property of multiplication: in order to multiply a number by a sum, you need to multiply it by each term separately.

Let's see how we actually do mental calculations.

Calculate:

Solution

1) Let's imagine how

2) Let's imagine the first factor as a sum of bit terms and perform the multiplication:

3) you can imagine how and perform multiplication:

4) Replace the first factor with an equivalent sum:

The distribution law can also be used in the opposite direction: .

Follow these steps:

1) 2)

Solution

1) For convenience, you can use the distributive law, only use it in the opposite direction - take the common factor out of brackets.

2) Let’s take the common factor out of brackets

Rice. 1. Illustration for the problem statement

Solution

Method 1. You can separately find out how much money it will take to buy linoleum for the kitchen, and then put it in the hallway and add up the resulting products.

An algebraic expression in which, along with the operations of addition, subtraction and multiplication, also uses division into letter expressions, is called a fractional algebraic expression. These are, for example, the expressions

We call an algebraic fraction an algebraic expression that has the form of a quotient of the division of two integer algebraic expressions (for example, monomials or polynomials). These are, for example, the expressions

The third of the expressions).

Identical transformations of fractional algebraic expressions are mostly aimed at representing them in the form of an algebraic fraction. To find the common denominator, factorization of the denominators of fractions is used - terms in order to find their least common multiple. When reducing algebraic fractions, the strict identity of expressions may be violated: it is necessary to exclude values of quantities at which the factor by which the reduction is made becomes zero.

Let us give examples of identical transformations of fractional algebraic expressions.

Example 1: Simplify an expression

All terms can be reduced to a common denominator (it is convenient to change the sign in the denominator of the last term and the sign in front of it):

Our expression is equal to one for all values except these values; it is undefined and reducing the fraction is illegal).

Example 2. Represent the expression as an algebraic fraction

Solution. The expression can be taken as a common denominator. We find sequentially:

Exercises

1. Find the values of algebraic expressions for the specified parameter values:

2. Factorize.

Section 5 EXPRESSIONS AND EQUATIONS

In this section you will learn:

ü o expressions and their simplifications;

ü what are the properties of equalities;

ü how to solve equations based on the properties of equalities;

ü what types of problems are solved using equations; what are perpendicular lines and how to build them;

ü what lines are called parallel and how to build them;

ü what is a coordinate plane?

ü how to determine the coordinates of a point on a plane;

ü what is a graph of the relationship between quantities and how to construct it;

ü how to apply the studied material in practice

§ 30. EXPRESSIONS AND THEIR SIMPLIFICATION

You already know what letter expressions are and know how to simplify them using the laws of addition and multiplication. For example, 2a ∙ (-4 b ) = -8 ab . In the resulting expression, the number -8 is called the coefficient of the expression.

Does the expression CD coefficient? So. It is equal to 1 because cd - 1 ∙ cd .

Recall that converting an expression with parentheses into an expression without parentheses is called expanding the parentheses. For example: 5(2x + 4) = 10x+ 20.

The reverse action in this example is to take the common factor out of brackets.

Terms containing the same letter factors are called similar terms. By taking the common factor out of brackets, similar terms are raised:

5x + y + 4 - 2x + 6 y - 9 =

= (5x - 2x) + (y + 6 y )+ (4 - 9) = = (5-2)* + (1 + 6)* y -5 =

B x+ 7y - 5.

Rules for opening parentheses

1. If there is a “+” sign in front of the brackets, then when opening the brackets, the signs of the terms in the brackets are preserved;

2. If there is a “-” sign in front of the brackets, then when the brackets are opened, the signs of the terms in the brackets change to the opposite.

Task 1. Simplify the expression:

1) 4x+(-7x + 5);

2) 15 y -(-8 + 7 y ).

Solutions. 1. Before the brackets there is a “+” sign, so when opening the brackets, the signs of all terms are preserved:

4x +(-7x + 5) = 4x - 7x + 5=-3x + 5.

2. Before the brackets there is a “-” sign, so when opening the brackets: the signs of all terms are reversed:

15 - (- 8 + 7y) = 15y + 8 - 7y = 8y +8.

To open the parentheses, use the distributive property of multiplication: a( b + c ) = ab + ac. If a > 0, then the signs of the terms b and with do not change. If a< 0, то знаки слагаемых b and change to the opposite.

Task 2. Simplify the expression:

1) 2(6 y -8) + 7 y ;

2)-5(2-5x) + 12.

Solutions. 1. The factor 2 in front of the brackets is positive, therefore, when opening the brackets, we preserve the signs of all terms: 2(6 y - 8) + 7 y = 12 y - 16 + 7 y =19 y -16.

2. The factor -5 in front of the brackets is negative, so when opening the brackets, we change the signs of all terms to the opposite:

5(2 - 5x) + 12 = -10 + 25x +12 = 2 + 25x.

Find out more

1. The word “sum” comes from Latin summa , which means “total”, “total amount”.

2. The word “plus” comes from Latin plus which means "more" and the word "minus" is from Latin minus What does "less" mean? The signs “+” and “-” are used to indicate the operations of addition and subtraction. These signs were introduced by the Czech scientist J. Widman in 1489 in the book “A quick and pleasant account for all merchants”(Fig. 138).

Rice. 138

REMEMBER THE IMPORTANT

1. What terms are called similar? How are similar terms constructed?

2. How do you open parentheses preceded by a “+” sign?

3. How do you open parentheses preceded by a “-” sign?

4. How do you open parentheses preceded by a positive factor?

5. How do you open parentheses that are preceded by a negative factor?

1374". Name the coefficient of the expression:

1)12 a; 3) -5.6 xy;

2)4 6; 4)-s.

1375". Name the terms that differ only by coefficient:

1) 10a + 76-26 + a; 3) 5 n + 5 m -4 n + 4;

2) bc -4 d - bc + 4 d ; 4)5x + 4y-x + y.

What are these terms called?

1376". Are there any similar terms in the expression:

1)11a+10a; 3)6 n + 15 n ; 5) 25r - 10r + 15r;

2) 14s-12; 4)12 m + m ; 6)8 k +10 k - n ?

1377". Is it necessary to change the signs of the terms in brackets, opening the brackets in the expression:

1)4 + (a+ 3 b); 2)-c +(5-d); 3) 16-(5 m -8 n)?

1378°. Simplify the expression and underline the coefficient:

1379°. Simplify the expression and underline the coefficient:

1380°. Combine similar terms:

1) 4a - Po + 6a - 2a; 4) 10 - 4 d - 12 + 4 d ;

2) 4 b - 5 b + 4 + 5 b ; 5) 5a - 12 b - 7a + 5 b;

3)-7 ang="EN-US">c+ 5-3 c + 2; 6) 14 n - 12 m -4 n -3 m.

1381°. Combine similar terms:

1) 6a - 5a + 8a -7a; 3) 5s + 4-2s-3s;

2)9 b +12-8-46; 4) -7 n + 8 m - 13 n - 3 m.

1382°. Take the common factor out of brackets:

1)1.2 a +1.2 b; 3) -3 n - 1.8 m; 5) -5 p + 2.5 k -0.5 t ;

2) 0.5 s + 5 d; 4) 1.2 n - 1.8 m; 6) -8r - 10k - 6t.

1383°. Take the common factor out of brackets:

1) 6a-12 b; 3) -1.8 n -3.6 m;

2) -0.2 s + 1 4 d ; A) 3p - 0.9 k + 2.7 t.

1384°. Open the brackets and combine similar terms;

1) 5 + (4a -4); 4) -(5 c - d) + (4 d + 5c);

2) 17x-(4x-5); 5) (n - m) - (-2 m - 3 n);

3) (76 - 4) - (46 + 2); 6) 7(-5x + y) - (-2y + 4x) + (x - 3y).

1385°. Open the brackets and combine similar terms:

1) 10a + (4 - 4a); 3) (s - 5 d) - (- d + 5c);

2) -(46- 10) + (4- 56); 4)-(5 n + m) + (-4 n + 8 m)-(2 m -5 n).

1386°. Open the brackets and find the meaning of the expression:

1)15+(-12+ 4,5); 3) (14,2-5)-(12,2-5);

2) 23-(5,3-4,7); 4) (-2,8 + 13)-(-5,6 + 2,8) + (2,8-13).

1387°. Open the brackets and find the meaning of the expression:

1) (14- 15,8)- (5,8 + 4);

2)-(18+22,2)+ (-12+ 22,2)-(5- 12).

1388°. Open parenthesis:

1)0.5 ∙ (a + 4); 4) (n - m) ∙ (-2.4 p);

2)-s ∙ (2.7-1.2 d ); 5)3 ∙ (-1.5 r + k - 0.2 t);

3) 1.6 ∙ (2 n + m); 6) (4.2 p - 3.5 k -6 t) ∙ (-2a).

1389°. Open parenthesis:

1) 2.2 ∙ (x-4); 3)(4 c - d )∙(-0.5 y );

2) -2 ∙ (1.2 n - m); 4)6- (-р + 0.3 k - 1.2 t).

1390. Simplify the expression:

1391. Simplify the expression:

1392. Combine similar terms:

1393. Combine similar terms:

1394. Simplify the expression:

1)2.8 - (0.5 a + 4) - 2.5 ∙ (2a - 6);

2) -12 ∙ (8 - 2, by ) + 4.5 ∙ (-6 y - 3.2);

4) (-12.8 m + 24.8 n) ∙ (-0.5)-(3.5 m -4.05 m) ∙ 2.

1395. Simplify the expression:

1396. Find the meaning of the expression;

1) 4-(0.2 a-3)-(5.8 a-16), if a = -5;

2) 2-(7-56)+ 156-3∙(26+ 5), if = -0.8;

m = 0.25, n = 5.7.

1397. Find the meaning of the expression:

1) -4∙ (i-2) + 2∙(6x - 1), if x = -0.25;

1398*. Find the error in the solution:

1)5- (a-2.4)-7 ∙ (-a+ 1.2) = 5a - 12-7a + 8.4 = -2a-3.6;

2) -4 ∙ (2.3 a - 6) + 4.2 ∙ (-6 - 3.5 a) = -9.2 a + 46 + 4.26 - 14.7 a = -5.5 a + 8.26.

1399*. Open the parentheses and simplify the expression:

1) 2ab - 3(6(4a - 1) - 6(6 - 10a)) + 76;

1400*. Arrange the parentheses to get the correct equality:

1)a-6-a + 6 = 2a; 2) a -2 b -2 a + b = 3 a -3 b .

1401*. Prove that for any numbers a and b if a > b , then the equality holds:

1) (a + b) + (a- b) = 2a; 2) (a + b) - (a - b) = 2 b.

Will this equality be correct if: a) a< b ; b) a = 6?

1402*. Prove that for any natural number a, the arithmetic mean of the previous and following numbers is equal to the number a.

PUT IT IN PRACTICE

1403. To prepare a fruit dessert for three people you need: 2 apples, 1 orange, 2 bananas and 1 kiwi. How to create a letter expression to determine the amount of fruit needed to prepare dessert for guests? Help Marin calculate how many fruits she needs to buy if: 1) 5 friends come to visit her; 2) 8 friends.

1404. Make a letter expression to determine the time required to complete your math homework if:

1) a min was spent on solving problems; 2) simplification of expressions is 2 times greater than for solving problems. How long did Vasilko spend on his homework if he spent 15 minutes solving the problems?

1405. Lunch in the school canteen consists of salad, borscht, cabbage rolls and compote. The cost of salad is 20%, borscht - 30%, cabbage rolls - 45%, compote - 5% of the total cost of the entire lunch. Write an expression to find the cost of lunch in the school canteen. How much does lunch cost if the price of salad is 2 UAH?

REVIEW PROBLEMS

1406. Solve the equation: