Math.

x = 3 Step 1: Apply the distributive law The first step is to expand the terms 5(4x + 3 ) and 4 (2x + 1 ) by distribution so 5(4x + 3 ), 4 (2x + 1 ) = 47 becomes 20x + 15, 8x, 4 = 47 Step 2: Combine like terms We can then collect like terms in: 20x, 8x + 15, 4 = 47 12x + 11 = 47 Step 3: Subtract 11 on both sides We then have to subtract 11 on both sides to isolate the term with x. 12x + 11, 11 = 47, 11 12x = 36 Step 4: Divide by 12 on both sides We then divide both sides of the equation by 12 to isolate x. (12x)/(12) = 36/12 1x = 3 We can drop the 1 on the left-hand side to x = 3 x = 3 is the solution of the equation.

x = 7 or -7 Step 1: Add 61 on both sides The first step in solving this problem is to add 61 from both sides of the equation. 4x², 61 + 61 = 135 + 61 You should then carry out the addition of the numbers on both sides of the equation. 4x² = 196 Step 2: Divide by 4 on both sides We then divide both sides of the equation by 4. (4x²)/(4) = 196/4 x² = 49 Step 3: Take the square root on both sides The final step is to take the square root on both sides to solve for x. x² = 49 √{x²} = √{49} x = 7 or, 7 If you are asking WHY the -7 gets included as well, it’s because any negative number squared, is positive. 7 × 7 = 49 But so does: -7 × -7 = 49 Since both options work, we have to include both.

$24.50 This is a proportion problem and the given information allows us to formulate the equation below: {5 tickets}{$17.50} = {7 tickets}{x} where x is the cost of 7 lottery tickets. We can solve this equation by first cross-multiplying. 5 tickets × x = 7 tickets × 17.50 $ Carrying out the product on the right-hand side, we get; 5 tickets × x = 122.5 tickets • $ We can solve for x by dividing both sides of the equation by 5 tickets. {5 tickets × x}{5 tickets} = {122.5 tickets • $}{5 tickets} Carrying out the division, we get a 1 on the left-hand side and a 24.5 on the right-hand side. 1x = 24.5 $ We can drop the 1 next to x, to get x = 24.5 $. It would cost Jordy $24.50 to buy 7 lottery tickets.

28 trains This is a proportion problem and the given information allows us to formulate the equation below: {42 trains}{18 hours} = {x}{12 hours} where x is the number of trains that pass near Emma’s house in 12 hours. We can solve this equation by first cross-multiplying. 18 hours × x = 42 trains × 12 hours Carrying out the product on the right-hand side, we get: 18 hours × x = 504 trains•hours We can solve for x by dividing both sides of the equation by 18 hours. {18 hours × x}{18 hours} = {504 trains•hours}{18 hours} Carrying out the division, we get a 1 on the left-hand side and a 28 on the right-hand side. 1x = 28 trains We can drop the 1 next to x, to get x = 28 trains. In 12 hours, 28 trains pass near Emma’s house.

x = 15 To find the unknown value in a ratio, we rewrite the ratios as fractions and use some conversions or cross-multiply. Step 1: Rewrite the ratios as fractions 5:12 = x:36 is equivalent to 5/12 = (x)/(36) Step 2: Get the same denominator on both sides Note that 36 = 12 × 3, so we can just multiply the numerator and denominator of the left-hand side fraction by 3 to get the same denominator. ((5 × 3))/((12 × 3)) = (x)/(36) 15/36 = (x)/(36) Step 3: Get rid of the denominator. Multiply both of the sides by 36. ( 15/36) { × 36 =} ( (x)/(36)) {× 36} x = 15

x = 27 To find the unknown value in a ratio, we rewrite the ratios as fractions and use some conversions or cross-multiply. Step 1: Rewrite the ratios as fractions 30:45 = 18:x is equivalent to ( 30/45) = ((18)/(x)) Step 2: Cross-multiply the fractions 30 × x = 18 × 45 30x = 810 Step 3: Divide both of the sides by 30 to get x ( (30x)/(30)) = (810/30) 810 ÷ 30 = 27 x = 27

x = 51 To find the unknown value in a ratio, we rewrite the ratios as fractions and use some conversions or cross-multiply. Step 1: Rewrite the ratios as fractions. 0.75:x = 3:204 is equivalent to ( (0.75)/(x) ) = (3/204) Step 2: Cross-multiply the fractions. 0.75 × 204 = 3 × x 153 = 3x Step 3: Divide both of the sides by 3 to get x. ( (3x)/(3) ) = ( 153/3) 153 ÷ 3 = 51 x = 51

0.55 There are two methods we can use to solve this problem. Method 1: Using Equivalent Fractions Notice that the denominator of the fraction is a factor of 100. We can multiply both the numerator and the denominator by 5 to get an equivalent fraction with a denominator of 100: 11/20 = (11 × 5)/(20 × 5) 55/100 55/100, or fifty-five hundredths, is written as 0.55 as a decimal. Method 2: Use a calculator to divide the numerator and denominator. The simplest way to work out the fraction is by just dividing 11 by 20 using a calculator: 11/20 { = 11 ÷ 20} = 0.55

262 boxes Step 1: Establish a match The expression “29 % less than on the second day” means that 100 %, 29 % = 71 % of the number of boxes collected on the second day were collected on the first day. Now, we know that 186 is 71 % of the number that we are looking for. Let this number be x. We will setup an equation for x and solve it. Step 2: Set up equalities First of all, we will define the relationship between values and percentages: 100 % = x and 71 % = 186 Step 3: Set up an equation We will divide the second equation by the first equation: 100/71 = (x)/(186) Remember that you can perform the division only if the corresponding sides of the equations have the same units (e.g. percents must be divided by percents). Step 4: Solve the equation for x 100/71 = (x)/(186) Finally, we will multiply both sides of the equation by 186: (x)/(186) × 186= 100/71 × 186 (186x)/(186) = (18,600)/(71) x = 18,600 ÷ 71 x = 261.97 Since we are rounding to the nearest whole number, the answer is 262. Therefore, 262 boxes were collected on the second day.

48 pints For this problem, we are going from bigger units, i.e. gallons, to smaller units, i.e. pints. We will first convert the gallon into quarts, and then convert the quarts into pints. To convert gallons into quarts, we multiply the number of gallons by 4 and to convert quarts into pints, we multiply the number of quarts by 2. 6 gallons × (4 quarts/gallon) × (2 pints/quart) = 48 pints Therefore, there are 48 pints in 6 gallons.

0.17235 Remember that kilo means “one thousand” so there are 1,000 grams in 1 kilogram. To convert grams (g) into kilograms (kg), we divide the number of grams (g) by 1,000. {172.35 g}{1,000} = 0.17235 kg 172.35 grams (g) are equal to 0.17235 kilograms (kg).

262.78 Make sure the numbers are written vertically with decimal points aligned: {cccccccccc} {1} {1} {1} 6 7. 9 8 + 1 9 4. 8 0 2 6 2. 7 8

16 days When Shawn takes 8 ounces of his medication, he decreases the contents of the bottle by 8 ounces. Then, he repeats it for the second time, then for the third, he will decrease the contents of the bottle by 8 ounces every time. We need to know, how many times 8 goes into 128 so we use division: {array}{r} 016 8 {longdiv}{128} -08 48 -48 0 128 ÷ 8 = 16 The answer is 16 days, with no medication remaining.

71.60 To answer this question, we must divide the total amount collected by 3 (the number of servers that worked that evening): Each person received $71.60 after the dinner shift.

40 miles Step 1: This month Jane walked a fraction of the distance she walked last month, 36 miles is part of a whole. To get the whole, you first have to turn the percentage into a decimal. 90 % = 90/100 = 0.9 Step 2: We then divide the part by the decimal in the last step to get the whole. (36)/(0.9) = 40 As a way of checking, you can divide 36 by 40 and take note of the decimal you get. 36/40 = 0.9, which tells us that our answer is correct. Jane walked 40 miles last month.

Step 1: Recall that 1 foot = 0.3048 meters. We can use this information to setup the following proportionality equation: (1 foot)/(0.3048 meters) = (x feet)/(200 meters) Step 2: We can then use cross-multiplication to solve for the missing term. (0.3048 meters) × (x feet) = (1 foot) × (200 meters) 0.3048 x = 200 feet Step 3: Finally, we can divide both sides of the equation by 0.3048 to solve for x: (0.3048 x)/(0.3048) = (200 feet)/(0.3048) x = 656.17 ft

5 This is a division problem, June is dividing the ½ liter into cups. We will first convert the mixed number into an improper fraction. We will then divide it by the total amount of juice by the capacity of each cup to arrive at our answer. Step 1: Convert the mixed number into an improper fraction We will first convert 2 ½ into an improper fraction: 2 ½ = (2 × 2 + 1)/(2) = 5/2 2 ½ is 5/2 as an improper fraction. Step 2: Set Up Your Problem Using KCF [Keep, Change, Flip] When we divide fractions we setup our problem: 5/2 ÷ ½ Then we follow KCF, Keep, Change, Flip. Keep: The first fraction, 5/2 stays the same. Change: The division sign changes to a multiplication sign Flip: The second fraction flips over to become 2/1 The new problem now reads: 5/2 × 2/1 Step 3: Cross Simplify [If Applicable] When we look at the fractions to cross-simplify, we find that 2 and 2 can be reduced. 5/2 × 2/1 = (5 × (2 ÷ 2 ))/((2 ÷ 2 ) × 1) (5 × 1)/(1 × 1) Step 4: Multiply When we multiply fractions, we multiply the numerators together and the denominators together: (5 × 1)/(1 × 1) = 5/1 = 5 Therefore, June can serve 5 full cups of orange juice.

3/22 Step 1: When we multiply fractions, we multiply all the numerators together and all the denominators together: 5/11 × 3/10 = (5 × 3)/(11 × 10) = 15/110 Step 2: We will simplify this fraction by dividing both the numerator and the denominator by the greatest common factor of 15 and 110. Factors of 15: 1, 3, 5, 15. Factors of 110: 1, 2, 5, 11, 22, 55, 110 Step 3: The greatest common factor of 15 and 110 is 5. We will divide both the numerator and the denominator by 5: 15/110 = (15 ÷ 5)/(110 ÷ 5) = 3/22

2,916 inches We are converting bigger units, i.e. yards, into smaller units, i.e. inches. First, we have to convert the yards into feet. To convert yards into feet, we multiply the number of yards by 3. 81 yards × ({3 feet}{yard} ) = 243 feet Now, we have to convert the feet into inches. To convert feet into inches, we multiply the number of feet by 12. 243 feet × ({12 inches}{foot} ) = 2,916 inches The model car travels a distance of 2,916 inches.

0.673 To convert from percent to decimal, you divide the percent by 100. 67.3 % = 67.3 ÷ 100 = 0.673 Note that converting from percent to decimals has the effect of moving the decimal point two spaces left.

4 ½ Step 1: Сonvert mixed numbers into improper fractions. We first convert each of these numbers into improper fractions. Any whole number, like 7, has a denominator of 1. 7 is 7/1 as an improper fraction. 2 ½ = (2 × 2 + 1)/(2) = 5/2 2 ½ is 5/2 as an improper fraction: Step 2: Subtract the fractions. We then take the difference between the two improper fractions. 7, 2 ½ = 7/1, 5/2 Step 3: Reduce to a common denominator. The denominators of the improper fractions are not the same, so we can not proceed with subtraction. We first have to find the LCM of 1 and 2 which is 2. We rewrite each of the fractions with a denominator of 2 and find the difference. 7/1, 5/2 = (7 × 2)/(1 × 2), (5 × 1)/(2 × 1) = 14/2, 5/2 = 9/2 Step 4: Convert the improper fraction into a mixed number. The last step is to convert the improper fraction into a mixed number. 9/2 {= 9 ÷ 2} = 4 ½

425 apples One box contains 25 apples, the second box contains another 25, the third another 25, and so on until 17. So, we need to add 25 to itself a total of 17 times to find the total number of apples. When we need to repeat the addition of 25 seventeen times, we simply can use multiplication: 25 × 17 {3} 2 5 × 1 7 1 7 5 2 5 0 4 2 5 25 × 17 = 425 There are 425 apples in 17 boxes.

2 5/14 Step 1: We will use the KCF method to work out the problem. We will keep 3/7 as it is, convert the division (÷) sign into a multiplication (×) sign, and then flip 2/11 to 11/2: 3/7 ÷ 2/11 = 3/7 × 11/2 Step 2: We will then multiply the two fractions by multiplying the numerators together and the denominators together: 3/7 ÷ 11/2 = (3 × 11)/(7 × 2) = 33/14 Step 3: Finally, we will convert the improper fraction into a mixed fraction: 33/14 {= 33 ÷ 14} = 2 5/14

178.6 liters Recall that 1 gallon = 3.8 liters. We can use this information to setup the following proportionality equation: 47 × 3.8 liters = 178.6 liters Therefore, the barrel contains 178 liters of gas.

31/50 Step 1: We first have to read the fraction correctly according to the decimal place value. The decimal 0.62 is sixty-two hundredths. Step 2: We create a fraction from how the decimal is read. The fraction for sixty-two hundredths is 62/100. Step 3: Since we do not see 62/100 as an answer choice it means we must simplify our fraction. We then have to divide both numerator and denominator of 62/100 by the same number until we get to a fraction in its simplest form. Since both numbers are even, we will first divide both numerator and denominator by 2. 62/100 = (62 ÷ 2)/(100 ÷ 2) = 31/50 0.62 is 31/50 as a fraction in its simplest form.