positive

* = “of” So, 1/3 * 36 = one-third of thirty-six which is the same as a third of thirty-six or the same as 36/3 because 1/3 * 36 = 1/3 * 36/1 cancel out the 1s and I’m left with just 36/3 So, 1/3 * 36 = 12

top number(numerator) goes in the division bracket and bottom number(denominator) goes outside to the left of the division bracket.

when asked whats the square root of something, im actually being asked “what” squared will give me 9. in this case, 3 squared(3²) will give me 9. So, the square root of 9 is 3.

find closest perfect square root below 87 √ 81 or 9² is closest to √ 87 So, put down 9 Subtract the two radicands 87-81=6. put 6 as numerator Double the square root 9*2=18. put 18 as denominator Combine the square root with the fraction 9⁶/₁₈ Solve to decimal form ⁶/₁₈ = 6 ÷ 18 = 0.33 Add that to the 9. Answer is 9.33

23

36

get rid of decimals and multiply 634 x 725 you get 460,650 count total digits to right of decimal places to right of decimal in “6.34” there’s “3” and “4” to right of decimal in “7.25” there’s “2” and “5” 3, 4, 2, and 5 is 4 toltal digits So, move decimal in 460,650 to the left 4 times 460,650.0 to left 4 times = 46.0650 6.34 x 7.25 = 46.0650

adding its opposite

67 + (-3.89)

subtracting its opposite

-5.87 - (-7³/₈)

(-4+4)+5, 5

-look at what numbers are positive and negative -in 9⁴/₇ + (-5.3), we have: -positive 9⁴/₇ and negative 5.3 -in -5.3 - (-9⁴/₇): -the “5.3” is negative and -the “- (-9⁴/₇)” makes the 9⁴/₇ a positive because -subtracting a number is the same as adding it’s opposite -So, - (-9⁴/₇) is same as + (9⁴/₇) -you can get rid the parenthesis cuz it’s essentially just + 9⁴/₇

commutative: 3 x 4 = 4 x 3 (no matter the order) associative: (2 x 4) x 3 = 2 x (3 x 4) no mater wat identity: 1 x 7 = 7 and 8 x 1 = 8

Commutative property: changing the order of factors does not change the product Associative property: chnaging the grouping of factors does not change the product Identity property: the product of 1 and any number is that number.

make a graph and plot the points. or do it in your head. B and C have the same y-axis but differnt x so their horizontal line runs across the x-axis between each other. A and B have the same x-axis but differnt y so their vertical like runs across the y-axis between each other. If this is a rectangle, A doesnt have a counterpart like B and C are. BC have same y but different x. so A and D must have same y but didferent x. B and C is 3 units long so A and D must be 3 units long. D is (8,1)

Since AB and DC are the same length and they’re both horizontal segments, this means that they run across the x-axis with unchanging y-axis. so we need to identify which points are horizontal from each other and subtract their “x” values. C and D have same y but x is defferent which mens it runs horizontally across the x-axis. the difference between the x of C and D is 10 units: 6.5 - (-3.5) = 6.5 + 3.5 = 10 So, since both AB and CD are the same length and bith are horizontal, just add 10 unis to the “x” in verticy A to find the coordinate of verticy B. 3 + 10 = 13. So, verticy B is (13,3)

find a common factor between 12 and 8 - 12*2=24 and 8*3=24 what every was multiplied by the initial denominator, the same must be done to the numerator - 7*2=14 and 6*3=18 then you’ll have the same denominator and numerator - ¹⁴/₂₄ ? ¹⁸/₂₄ which ever numerator is the bigger is the fraction thats is greater than the otter fraction. - So, ⁴/₂₄ < ¹⁸/₂₄

⁷/₂

the “₄” tells us how many pieces are in a whole for the fraction 5¹/₄. imagine theres 5 pies with 4 pieces in each pie. Then you have an extra ¹/₄ of a piece or just one slice added to the 5 pies So just add all of tye pieces in total. 5*4+1=21 So, the impropper fraction for 5¹/₄ is ²¹/₄

do the opposite of what you do when you’re turning a mixed number into an improper fraction. instead of multiply first, you divide. - 29/5=5 with an extra 4 remaining So, instead of making the remainder a decimal just make it a fraction by putting it as the numerator over “₅” - 5⁴/₅

turn them into improper fractions first for 4¹/₅ - 5 pieces * 4 plus an extra 1 = ²¹/₅ for 1²/₅ - 5 pieces * 1 plus an extra 2= ⁷/₅ Since denominators are the same, just add the numerators. - ²¹/₅ + ⁷/₅ = ²¹+⁷ = ²⁸/₅

turn them into improper fractions first for 6⁷/₄ - 4 pieces * 6 plus an extra 7 = ³¹/₄ for 5³/₈ - 8 pieces * 5 plus an extra 3= ⁴³/₈ Since denominators are not the same, find closest common factor between denominators 4 and 8 are in the same multiples of 4 - 4, 8, 12, 16, and so on… the first number they have in common is 8 So, the ⁴³/₈ can stay the same since the denominator is already 8 However, for ³¹/₄, we need to multiply the denominator, “₄” by 2 to make it “₈” as well as the numerator, “³¹” by 2 to balance it out. whatever you do on the denominator you muat do to the numerator. - So ³¹/₄ x2 = ⁶²/₈ Now we have ⁶²/₈ - ⁴³/₈ = ²¹/₈

²/₃ x 3 is basically just saying ²/₃ + ²/₃ + ²/₃ which, if you know how to add fractions, you know that you dont need to do anything with the denominators unless they arent the same. So, ²/₃ + ²/₃ + ²/₃ or ²*³/₃ = ⁶/₃

find the reciprocal of the second or first fraction - reciprocal: ⁴/₉ = ⁹/₄ multiply the numerators together and denominators together - ²*⁹/₇*₄ = ¹⁸/₂₈ simplify that answer. 2 can go into both numerator and denominator - So 18÷2=9 and 28÷2=14 - ⁹/₁₄

Multiply the numerators together and then the denominators together - ³*⁸/₈*₉ = ²⁴/₇₂ Simplify the fraction. 24 can go into 72 3 times and into 24 1 time obviously. - 24/24=1 and 72/24=3 - ¹/₃

7 is the whole number in this fraction so the answer i going to be “7.” somthing. To find out we need to do 4/23. 23 cant go into 4 but it can go into 40. So in long division, you put a decimal behing “4” and then a “0” afterthat. 23 goes into 40 1 time with “17” left over. 23 cant go into 17 but it can go into 170. since we’re already behind the decimal, you dont put another decimal, just add a 0 begind the 17 and 23 goes into 170 7 times with 9 left over. Since we have .17 so far and the “7” is at the hundredths place, we can stop there. - The answer is: 7.17

7/19. 19 cant go into 7 but it can go into 70. So in long division, you put a decimal behing "7" and then a "0" afterthat. 19 goes into 70 3 times with "13" left over. 19 cant go into 13 but it can go into 130. since we're already behind the decimal, you dont put another decimal, just add a 0 begind the 13 and 23 goes into 130 6 times with 16 left over. to stop at hundredths place, we need to make sure the number in the thousandths place doesnt make the number in the hundrethds place round up. So 19 cant go into 16 but it can go into 160 8 times with 8 left over. So far we have .368 but the problem says round to nearest hundredth so the “8” makes the “6” go up a number. So, - The answer is 0.37

the decimal stops at the hundredths the place. So whatever, the fraction version of 0.78 is, its something over 100. the “.78” tells us the number that is iver 100. if the fraction was 100/100 then the number wouldnt be 0.78, it’d actually be 1. so since its ⁷⁸/₁₀₀, its saying 78 out if 100%. Now, we need to simplify the fraction ⁷⁸/₁₀₀. 2 can go into both 78 and 100. So, 78/2=39 and 100/2=50. Answer is: ³⁹/₅₀

the “3” in the ones places tells us this will be a mixed number because the whole number is always in the ones place when turning a mixed number into a decimal. So the answer is 3 and ?/?. the “2” is in the tens place so the fraction is something over 10, the 2 tells us whats over 10. So, we have 3²/₁₀. That can be simplified because 2 can go into bitg 2 and 10. 2/2=1 and 10/2=5 So, Answer is 3¹/₅

m - 10

6 - k

¹/₃ * r

10 * (2+3)

9x + 6

coefficients are 2 and 7 becaue they’re multiplied by a variable constants are the 1 and 5 becauee they arent being manipulated by a variable and they’re standalone/unchanging quantities.

2(3x + 5) is just saying there are two “3x + 5”s So: 3x + 5 +3x + 5 ————- 6x + 10 OR: You can use distributive property and do: 2 * 3x + 2 * 5 which is still 6x + 10

We can use distributuve property in both sides 7 * 3y = 21y and 7 * -5 = 35 so, its “21y - 35” -2 * 10 = -20 and -2 * 4y = -8y so, its “-20 - 8y” Now we have 21y - 35 - 20 - 8y We can still combine more like terms 21y and -8y is the same as 21y - 8y which is 13y -35 and -20 is the same as -35 - 20 which is -55 So, final answer is: 13y - 55

3 * 6 = 18 and 3 * -5 = -15 Is: 18 - 15 Which is: 3

4y

-3y + 2y = -y “and” +4xy - 4xy = 0xy “and” -2x² + 3x² = x² “and” +2x “and” +y² Answer = -y + x² + 2x + y²

32 = 2 x 16, 2 x 8, 2 x 4, 2 x 2 24 = 2 x 12, 2 x 6, 2 x 3 Both list of factors have 3 2s in common. So, 2 x 2 x 2 = 8 GCF = 8

148 = 2 x 74, 2 x 37 96 = 2 x 48, 2 x 24, 2 x 12, 2 x 6, 2 x 3 They both have 2 2s in common So, 2 x 2 = 4 GCF = 4

16: 2 x 8, 2 x 4, 2 x2 20: 2 x 10, 2 x 5 They both have 2 2s in list of factors So, 2 x 2 = 4 GCF = 4 Now turn it into distributive property GCF goes outside of group 4(?g + ?h) 4 x 4 = 16 So, 4(4g + ?h) 4 x 5 = 20 So, 4(4g + 5h) Answer: 4(4g + 5h)

27: 3 × 9, 3 x 3 18: 3 × 6, 2 × 3 They both 2 3s in common So, 3 x 3 = 9 Now turn into distributive property GCF goes outside of group 9(? + ?r) 9 x 3 = 27 So, 9(3 + ?r) 9 x 2 = 18 So, 9(3 + 2r) Answer: 9(3 + 2r)

16a + 20, 12a + 20 + 4a, 2(8a + 10)

3x

a combination of terms that are all linked together by addition or subtraction.

3x = monomial 3x + 67y = binomial 5t - 61xy + 6m = trinomial 9y - 43g + 51xg + 75x = polynomial

the value of the exponent on the variable.

This is a 5th degree because the degree of a polynomial is determined by the term with the highest exponent

5th degree because the variable in a term with multiple exponents have to add up the exponents to reveal or identify its true degree

from highest degree to smallest degree 2y⁴ + 3yx³ + 4x²

can be seen as a term because any number raised to the 0th power is just one. So, in the same way the variable “y” can be seen as 1y or 1 * y and you still get “y”, “2” can be seen as 2 * 1 or 2x⁰ since no matter what “x” is, if its raised to the 0th power, ots just going to equal 1. thats a fundamental rule mathematics.

Finding the domain of an expression involves identifying all possible values of the independent variable that allow the expression to be defined without causing mathematical inconsistencies or errors.

All real numbers except x ≠ -2, 4 because the denominator would equal zero if -2 or 4 was substituted for “x”.

the blocks on top mean addition. t+t+t+t+t+t. t+t+t+t+t+t is the same as t * 6 or t6 So, an equation for the diagram can be: t * 6 = 9

the blocks mean the sum of two y’s equal 7. y + y. y + y is the same as y * 2 or 2y So, 2y = 7

Subtract 1/3 on both sides to rearange the equation while keeping it balanced 1/3 - 1/3 + a = 5/4 - 1/3 the 1/3s cancwl out and we’re left with: a = 5/4 - 1/3 So, since the denominators arent the same we need to find the LCF between the two denominators and make then the same. factors of 3 are 3 6 9 12 15… factors of 4 are 4 8 12 16… They both have 12 in common and it’s the LCF. So, we multiply the denominator, “₃” by 4 to get 12 and we multiply the denomintor, “₄” by 3 to get 12 Now have a = 5/12 - 1/12 but whatever we do to the bottom we need to do to the top So, for the 5/12, we need to multiply 5 by 3 because thats what we did to the initial denominator, “₄”. 5 * 3 = 15 We need to do the same for the other fravtion, 1/12. the initial denominator was 3 and he had to multiply it by 4 to get 12 si we need to do the same for the numerator, “¹” to balance it. 1 * 4 = 4 So now we have: a = 15/12 - 4/12. we can finally subtract tye numerators since the denominators are the same. 15-4 = 11 So, Answer: a = 11/12 or ¹/₃ + ¹¹/₁₂ = ⁵/₄

Subtract 7/9 on both sides to rearange the equation while keeping it balanced 11/6 - 7/9 = n + 7/9 - 7/9 the 7/9s cancwl out and we're left with: 11/6 - 7/9 = n So, since the denominators arent the same we need to find the LCF between the two denominators and make then the same. factors of 6 are 6 12 18 24… factors of 9 are 9 18 27… They both have 18 in common and it's the LCF. So, we multiply the 1st denominator, "6" by 3 to get 18 and we multiply the 2nd denomintor, "9" by 2 to get 18 Now have 11/18 - 7/18 = n. But whatever we do to the bottom we need to do to the top So, for the 11/18, we need to multiply 11 by 3 because thats what we did to the initial denominator, "6". 11 * 3 = 33 We need to do the same for the other fraction, 7/18. the initial denominator was 9 and we had to multiply it by 2 to get 18. So, we need to do the same for the numerator, "⁷” to balance it. 7 * 2 = 14 So now we have: 33/18 - 14/18 = n we can finally subtract the numerators since the denominators are the same. 33 - 14 = 19 So, Answer: a = 19/18 or ¹¹/₆ = ¹⁹/₁₈ + ⁷/₉

this is the same as: d/4 = 9/1 something goes into “d” 9 times for the numerator when simplified and something goes into 4 1 times for the denominator when simplified. We know for the denominator it’ll be 4 because 4 goes into 4 just once. For goes into 36 9 times because: 9 * 4 = 36 So, ³⁶/₄ = ⁹/₁. Answer: d = 36

First off,2/3p = 5 is the same as: ²/₃p = ⁵/₁ We can rearrange the equation and isolate “p” by dividing both sides both sides by ²/₃. which is the opposite of multiplication. ²/₃ ÷ ²/₃p = ⁵/₁ ÷ ²/₃ We know when dividing fractions, we’re supposed to find the reciprocal of one of the fraction and seitch the division to multiplication. ³/₂ x ²/₃p = ⁵/₁ x ³/₂ the left side is equal to 1p or just “p” because: when multipkyyng fractions we multiply the numers together and then the denominators togther: ³*²/₂*₃ = ⁶/₆ or just 1 So we have: p = ⁵/₁ x ³/₂ multiply numeratirs and denominatirs together: ⁵*³/₁*₂ = ¹⁵/₂. No whole number can go into 15 and 2 equally. So this is already simplified There for, p = ¹⁵/₂ or ²/₃ * ¹⁵/₂ = ³⁰/₆ = 5

g/₄ = 3.2 is the same as: g ÷ 4 = 3.2 we can isolate “g” and rearange the equation by multiplying 4 on both sides. doing it to both sides balances the equation. 4 * g ÷ 4 = 3.2 * 4 The 4s on the left side cancel out becauee multiplication is the inverse of division and we’re left with: g = 3.2 * 4 Which is: g = 12.8.

-8 < x

D

x ≤ -4

No, the circle should be filled in if x is also “equal” to 3.

An inequality in math is a statement that compares two values or expressions using inequality symbols. It shows that one value is either <, >, ≤, ≥, or ≠ to another value.

3 ≥ x

j < 14

t > 100

independent variable is the variable that can be easily changed at anytime dependant variable is the variable that we’re solving for or the variable whos value is literally depending on the value of the independent variable. G is the dependent variable because G will change depending on what “a” is. For ex: if “a=2” then “G = 10”. “a” can be changed out anytime so its the independent variable.

obviously 1 * 5 = 5. So, x = 1 in the first box. in the second box, we can plug in 4 for y since y = 4. Then you’d have 4 = 5x. We can isolate “x” by dividing both sides by 5. 5 divided by 5x will cancel out the 5’s on the right side, and on the left side we have: 5 ÷ 4 or ⁵/₄ So ⁵/₄ = x in the third box, we do the same thing we did for the second box. plug in 2 and then divide bith sides by 5 to get 5 ÷ 2 = 5x ÷ 5. Cancel out 5’s in the right and you’re left with: 5 ÷ 2 = x or ⁵/₂ = x There for: x = 1 if y = 5 x = ⁵/₄ if y = 4 x = ⁵/₂ if y = 2

first box, we’re given the x. plug it in and you get ⁸/₂ which is just 8 ÷ 2. which equals. So y = 4 second box, we’re given the x. plug it in and you get ¹⁴/₂ which is just 14 ÷ 2. which equals 7. So y = 7 third box, we’re goven the y. plug it in and you get 3 = ˣ/₂ or 3 = x ÷ 2. we can isolate x by multiplying both sides by 2: 2 * 3 = x ÷ 2 * 2. the 2’s cancel out on the right side and we’re left with: 2 * 3 = x. So x = 6

To find the answer for these match equation to coordinates problems, just look at the coordinates (x, y), of each point on the graph and plug them into the equation “y = 0.5x + 5” start with the first graph on the left. the first coord on this graph is (0, 5). plug them into the equation and you get: 5 = 0.5 * 0 + 5 which is right so far. plugging in the second coord will be: 2 = 0.5 * 4 + 5. when solved it’ll be 2 = 7. which is incorrect. So the first graph is inconsistent. Lets try the second graph in the middle. the first coord on this graph is (1, 0). plug them into the equation and you get: 0 = 0.5 * 1 + 5. when solved it’ll be 0 = 1. which is incorrect. So the second graoh is already off. in the last graph on the right, the first coord on this graph is (2, 6). plug them into the equation and you get: 6 = 0.5 * 2 + 5 which is right so far(6=6). the second coord on this graph is (4, 7). plug them into the equation and you get: 7 = 0.5 * 4 + 5 which is right so far(7=7). the third coord on this graph is (6, 8). plug them into the equation and you get: 8 = 0.5 * 6 + 5 which is right so far(8=8). So the last graph on the right is completely consistent when plugging in all tye coords.

lets look at the forst rule "y = x - 1" and plug in the first coords (2, 3). "3 = 2 - 1" is not true. once you find one untrue statement, then you know that the rule or expression is incorrectly representing the graph. So move on to the second rule "y = x + 1" and plug in the first coords (2, 3). "3 = 2 + 1" which is true. plugging in seconds coords will be: "6 = 5 + 1" which is also true. plugging in third coords will be: "8 = 7 + 1" which is true. the second rule/expression is thoroughly consistent with respeesenring the graph.

1. When you see the value of a variable(l) “based” on another variable(w), you know that “l” is going to be the the result or dependent variable and “w” is going to be the independent varible. So you can start off with l = w. but we need to write an equation that represents the length based on its width. if the garden is 24 squate meters, then that means its length times its width will yeild 24. l * w = 24. if we want the length to be based or dependent on the width, then we need to rearange this equation to make “l” the dependent variable and “w” the independent variable. We can divide bothe sides by “w” in order tk isolate “l”. w ÷ l * w = 24 ÷ w The w’s cancel out in tye left and we’re left with: l = 24 ÷ w or l = ²⁴/w 2. To find the legnth when we’re given the width. just plug in the the 2.5 in for the width and you get: l = ²⁴/2.5. 24 ÷ 2.5 = 9.6 So l = 9.6

it says to write an equation that represents the amount of discount offered(d), that lets me know that “d” is going to be the dependent variable because whenever u represent something that means you’re making it the point of interest. So we can start off with d = write an equation that represents the amount of discount offered(d) on an item whose usual price is p. we’re given that the discount is 15%. if the usual price is “p”, then we need to do somethjng to “p” that will make “d” equal or represent the amount that was dicounted. so we need to write “15% of p” in the form of a mathematical expression. 15% is the same as ¹⁵/₁₀₀ and “of” is the same as “*”. So we can write: ¹⁵/₁₀₀ * p or just ¹⁵/₁₀₀p So now we’ve represented “d”(d =) and we’ve made it equal the amount discounted off of an item from its usual price “p”(¹⁵/₁₀₀p) There for: d = ¹⁵/₁₀₀

“Write an equation that represents the distance Jamie will run in kilometers(d) at a rate of r kilometers per hour”. I can read that sentence and substitute "r" for an actual value and then solve it to help me understand what equation l'm supposed to write. For example, if i substitute 10 in for "r", the question reads: "Write an equation that represents the distance Jamie will run in kilometers(d) at a rate of 10 kilometers per hour" If Jamie will be running for ¹/₂ hours, in one hour she would have covered 10 kilometers, so in half an hour she wiuld have covered half that distance, 5 kilometers. Think about how the wuesyions says “represemt the distance Jamie will run in kilometers(d)” That lets me know that “d” will be the point of interest. So, i can start with d = Think what I need to do to “r” kilometers, or in this case 10 kilometers in order to turn it into 5 kilometers. Since we are given ¹/₂ or 0.5 as the time spent running, we can multiply ¹/₂ or 0.5 by 10 to get 5 kilometers. So we have: d = ¹/₂r

There are currenty constants and variable terms on each side of the equation. We need to start by tryna get variable terms on one side of the equation and the constant on the other side of the equation. subtract 8 on bith sides. 8’s cancel out on left side and we left with: -4s = s + 5 the right side has variable term and constant in it, so since the left only has a variabke term in it, its best to bring the “s” variabke to the legt side with “4s” on order to get variable terms on one side of the equation and the constant on the other sode of the equation. So since “s” is the same as “1s”, we can subtract 1s on bith sides. the “s”’s on the right side cancel out and we’re ledt with: -5s = 5 Now we can isolate “s” on the left side by dividing both sides by “-5”. the “-5”’s on the keft side cancel out and we’re left with: -5 ÷ -5s = 5 ÷ -5 s = -1

There are currenty constants and variable terms on each side of the equation. We need to do whatever we can to get variable terms on one side of the equation and the constants on the other side of the equation. we can isolate terms, remove or combine terms to achieve this. first we should start by combining like terms to make it easier to start organizing/rearranging variables terms to one side and constants on the other. -f + 4f + 2 = 8 - 3f after combining like terms will be: 3f + 2 = 8 - 3f Now we can start puttin variable terms on one side and constants on the other. lets start by isolating the “8” on the right side by removing the “-3f”. we add 3f on both sides to cancel out the “-3f” and make the “3f” on the left side a “6f”. now we have: 6f + 2 = 8 the right side has only constant(s) on it but the left side is mixed, so we need to isolate the “6f” by removing the “2”. to do that, we subtract both sides by 2. the “2” on the left side cancels out and “8 - 2 = 6” so now we have: 6f = 6 Now in order to solve for “f” like the problem tells us to, we can just isolate the “f” on the left side. we do this by doing the inverse like we’ve been doing, the inverse of multiplication is division so we divide both sides by 6. the “6” on the left side cancels out and the 6 on the right turns into 1 because 6 ÷ 6 = 1. now we’re left with: f = 1

12:7

2:5

In a proportional relationship, the ratio between the two quantities is always the same. Is the ratio of kiwis left to number of days always the same? Let's make a table to see the number of kiwis left after different numbers of days. on day 1 = 73 kiwis on day 2 = 71 kiwis on day 3 = 69 kiwis the ratio between 73 and 1 is 73/1 = 73 the ratio between 71 and 2 is 71/2 = 35¹/₂ the ratio between 69 and 3 is 69/3 = 23 The ratios between the two quantities cannot be simplified to the same number, so the quantities are not proportional.

A relationship between two quantities is proportional if the ratio between those quantities is always equivalent. For each value of x, let's write the relationship of y to x as a ratio in simplest form. 2/⁵/₆ = 2 ²/₅ 10/²⁵/₆ = 2 ²/₅ 50/¹²⁵/₆ = 2 ²/₅ All of the ratios between the two quantities can be simplified to 2 ²/₅, so the quantities are proportional.

A scaling factor is a number that you multiply by another number(initial number) to scale it (increase or decrease it) to a desired value. For example: to scale 5 to 10, you multiply 5, the initial number, by 2, the scaling factor, in order to turn or scale 5 to 10. to scale 10 to 5, you multiply 10, the initial number, by 0.5, the scaling factor, in order to turn or scale 10 to 5.

the scaling factors for any fraction are the number you multiply the denominator by to get to the numerator and vise versa. For example: in ³/₁₁. the first scaling factor is the number that you multiply the denominator, “11”, by to scale it down to 3 is ³/₁₁ because ³/₁₁ = 0.2727… and 0.2727 * 11 = 3 the second scaling factor is the number that you multiply the numerator, “3”, by to scale it up to 11 is ¹¹/₃ because ¹¹/₃ = 3.6666… and 3.666 * 3 = 11

Since both sides are equal and all fractions have a scaling factor, if you know the scaling factor for one side, then its the same exact scaling factor for the other side. in this case we have the written out proportion of “³/k = ⁴/₅” and we need to find “k”. the denominator is missing or hnknown on the left side but the right side has indentifable denominator and numeratkr so lets start with the right side, “⁴/₅” we need to find the scaling factor of ⁴/₅ that scales the numerator to the denominator in order to find the correct scaling factor for the left side since its denominator is missing. in ⁴/₅, the scaling factor for the denominator is the number that you multiply the numerator, “4”, by to scale it to the denominator, “5”, is ⁵/₄ because ⁵/₄ = 1.25 and 1.25 * 4 = 5 So to find the denominator on the left side of the proportion, we use the same scaling favtor we used to find the denominator on the right side. So, 3 * ⁵/₄ or ³/₁ * ⁵/₄ = ¹⁵/₄. therefore, k = “¹⁵/₄” or 3.75

Since both sides are equal and all fractions have a scaling factor, if you know the scaling factor for one side, then its the same exact scaling factor for the other side. in this case we have the written out proportion of “r/₅ = ⁴/₇” and we need to find “r”. the numerator is missing or unknown on the left side but the right side has indentifiable numerator and denominator. So lets start with the right side, “⁴/₇” we need to find the scaling factor of ⁴/₇ that scales the denominator to the numerator in order to find the correct scaling factor for the left side since its numerator is missing. in ⁴/₇, the scaling factor for the numerator is the number that you multiply the denominator, “7”, by to scale it to the numerator, “4”, is ⁴/₇ because ⁴/₇ = 0.571428 and 0.571428 * 7 = 4 So to find the numerator on the left side of the proportion, we use the same scaling factor we used to find the numerator on the right side. So, 5 * ⁴/₇ or ⁵/₁ * ⁴/₇ = ²⁰/₇ therefore, r = “²⁰/₇” or 2.85714

A proportion in mathematics is a statement that two ratios are equal. For example, if you have two fractions or ratios, like a/b and c/d, a proportion would state that a/b = c/d. It shows that the relationship between the first pair of numbers is the same as the relationship between the second pair of numbers.

The problem says “She can keep the fire burning for 4 hours with 6 logs”. thats the same as saying 4 hours per 6 logs or ⁴/₆. Time over logs. you wana find how many logs it’ll take to burn for 18 hours. Since this is a proportion problem, you know that the relationship between the first fraction is the same as the relationship between the second fraction. to find the second fraction, since its time over logs and you wana find how many logs it’ll take to burn for 18 hours, you’ve been given the time, now you just gotta find the logs. so write that relationship the same way you wrote the 4 hoirs per 6 logs, just the variable “y” for logs. 18 hours per “y” logs or ¹⁸/y put it all together for a proportion statemnt and you have: ⁴/₆ = ¹⁸/y Now you just solve it the same way you would solve proportions. what times 4 will guve you 4? ⁶/₄ * 4 = 6. So you since the relationship between ⁴/₆ = ¹⁸/y are the same, you multiply 18 by ⁶/₄ as well. 18 * ⁶/₄ or ¹⁸/₁ * ⁶/₄ = ¹⁰⁸/₄. Simplified will give 27 logs needed to burn for 18 hours.

The problem says “She drove her car 99 kilometers and used 9 liters of fuel”. thats the same as saying 99 kilometers per 9 fuel or ⁹⁹/₉. Distance over fuel. you wana find how far she’ll go if she had 12 liters of fuel. Since this is a proportion problem, you know that the relationship between the first fraction is the same as the relationship between the second fraction. to find the second fraction, since its distance over fuel and you wana find how much distance she’ll cover if she had 12 liters of fuel, you’ve been given the fuel, now you just gotta find the distance(kilometers). so write that relationship the same way you wrote the 99 kilometers per 9 fuel, just the variable “k” for kilometers. “k” distance per 12 liters of fuel or k/₁₂ put it all together for a proportion statement and you have: ⁹⁹/₉ = k/₁₂ Now you just solve it the same way you always solve proportions. what times 9 will give you 99? 11 * 9 = 99. So, since the relationship between ⁹⁹/₉ = k/₁₂ are the same, you multiply 12 by 11 as well. 12 * 11 or ¹²/₁ * ⁹⁹/₉ = ¹¹⁸⁸/₉ Simplified will give 132 kilometers covered if she had 12 liters of fuel.

percent can be seen as. per - cent “per” means a division bar to denote something over something and “cent” means 100, as in “century”. So per - cent means something over 100 or a quantity for every 100 units of something.

if the entire square is one whole then you know that the entire square represents 100 percent. however, only a portion of the wntire square is shaded blue. to know what percentage is shaded, you count how many square there are in total and put that as tye denominator. then count how many are shaded and put that as the numerator. there are 100 squate in total so we have ?/₁₀₀ there are 92 squares shaded so now we have ⁹²/₁₀₀ solve the fraction and you get 0.92. you’re trying to say that something of 100 is shaded. that “something” is 0.92 of 100. “of” = “*” so just multiply 0.92 by 100 to find the per - cent and you get 92%.

if the each large square is one whole each then you know that the each square represents 100 percent. So both squares together would represent 200%. however, only a portion of the second square is shaded red. Since the two squares represent 100% each and the first swuare is completely shaded, that means the percent that’s represented by the shaded areas is at least 100%. to know what actual total percentage is shaded, you need to find the shaded percentage of the second swuare and add that to the first square, “100%” So, you count how many total squares there are in the second square and put that as the denominator. then count how many are shaded and put that as the numerator. there are 25 square in total so we have ?/₂₅ there are 13 squares shaded so now we have ¹³/₂₅ solve the fraction and you get 0.52. in percentage terms, you’re trying to say that something of 100 is shaded. that “something” is 0.52 of 100. “of” = “*” so just multiply 0.52 by 100 to find the per - cent and you get 52%. Now, add that 52% from the second square to 100% from the first square and you get 152%

you’re trying to find the percent decrease in the mass of the pumpkin. to do that you subtract how much he carved from what he started with and whatever that is, you divide that by what he started with and convert it to a percentage. he started with 6.5kg but he carved “x” of it and was left with 3.9kg. to find “x” you just simply subtract 3.9 from 6.5 which give you 2.6. Now you need to find the percent decrease in the mass. you know the mass decrease in kg, 2.6kg, but now you need to know what percentage of the original mass, “6.5”, is 2.6. l to do that, you simply do what you always do to find the percent “of” something. put the whole, in this case, the original mass as the denominator and the decrease in mass as the numerator. that will be: 2.6/6.5 = 0.4 convert that to a percent by multiplying 0.4 by 100 and you get 40% of the pumpkin was carved.

something that takes an input, something like “x” as the input and then the function does somethjng to it and then it spits out some other value called the output. which is going to be equal to “y”.

f(x)

all h(0) means is, when you input 0 into the function, “h”, what will be the output? in this case, what “y” will the function, “h”, be spitting out? another way to think about is relating it to function notation “f(x)”. when “x” is equal to 0, what will “y” be? based on the graph, when x == to 0, y == 5 so h(0) = 5