Cross-multiplying,When two ratios are equivalent, we say that they are proportional to each other. In the proportion equation, the ratio on the left side of the equation is equivalent to the ratio on the right side of the equation.
Consider an example where A : B is 50 : 100 and C : D is 30 : 60.
Reducing the ratios to lowest terms, we obtain the ratio of A : B as 1 : 2 and the ratio of C : D as 1 : 2.
Since these ratios are equivalent, they are proportional to each other, and their proportion equation is:
[latex] \boldsymbol{A : B = C : D} [/latex]
The proportion equation can also be formed by representing the ratios as fractions.
Equating the fraction obtained by dividing the first term by the second term on the left side, to the one obtained by dividing the first term by the second term on the right side, we obtain:
[latex] \boldsymbol{\displaystyle{\frac{A}{B} = \frac{C}{D}}} [/latex]
This proportion equation can be simplified by multiplying both sides of the equation by the product of both denominators, which is B × D.
[latex] \displaystyle{\frac{A}{B} = \frac{C}{D}} [/latex] Multiplying both sides by (B × D),
[latex] \displaystyle{\frac{A}{B}(B \times D) = \frac{C}{D}(B \times D)} [/latex] Simplifying,
[latex] AD = BC [/latex]
The same result can be obtained by equating the product of the numerator of the first ratio and the denominator of the second ratio with the product of the denominator of the first ratio and the numerator of the second ratio. This is referred to as cross-multiplication and is shown below:
Cross-multiplying,
[latex] AD = BC [/latex]
Notice that the cross-multiplication of [latex] \displaystyle{\frac{A}{C} = \frac{B}{D}} [/latex] also gives the same result of [latex] AD = BC [/latex]. Therefore, [latex] A : B = C : D [/latex] is equivalent to both [latex] \displaystyle{\frac{A}{B} = \frac{C}{D}} [/latex] and [latex] \displaystyle{\frac{A}{C} = \frac{B}{D}} [/latex]. If any three terms of the proportion equation are known, the fourth term can be calculated.
Consider the proportion equation [latex]A : B : C = D : E : F [/latex].
This proportion equation can also be expressed using fractions: [latex] \displaystyle{\frac{A}{B} = \frac{D}{E}} [/latex], [latex] \displaystyle{\frac{B}{C} = \frac{E}{F}} [/latex], and [latex] \displaystyle{\frac{A}{C} = \frac{D}{F}} [/latex]
Cross-multiplying leads to: [latex] AE = BD [/latex], [latex] BF = CE [/latex], and [latex] AF = CD [/latex]
The proportion [latex] A : B : C = D : E : F [/latex] can be illustrated by the table,
| 1st Term | 2nd Term | 3rd Term |
|---|---|---|
| A | B | C |
| D | E | F |
and expressed as, [latex] A : D = B : E = C : F [/latex]
Expressing using fractions, [latex] \displaystyle{\frac{A}{D} = \frac{B}{E} = \frac{C}{F}} [/latex]
Cross-multiplying leads to the same result as shown above: [latex] AE = BD, BF = CE, and AF = CD [/latex]
Therefore, [latex] A : B : C = D : E : F [/latex] is equivalent to both
[latex] \displaystyle{\frac{A}{B} = \frac{D}{E}, \frac{B}{C} = \frac{E}{F}, \frac{A}{C} = \frac{D}{F}} [/latex], and
[latex] \displaystyle{\frac{A}{D} = \frac{B}{E} = \frac{C}{F}} [/latex]
Determine the missing term in the following proportions:
[latex] 4 : 5 = 8 : x [/latex]
[latex] 6 : x = 10 : 25 [/latex]
[latex] x : 1.9 = 2.6 : 9.88 [/latex]
[latex] \displaystyle{3 : 3\frac{3}{4} = x : 5\frac{1}{4}} [/latex]
[latex] 4 : 5 = 8 : x [/latex]
Using fractional notation, [latex] \displaystyle{\frac{4}{5} = \frac{8}{x}} [/latex] or [latex] \displaystyle{\frac{4}{8} = \frac{5}{x}} [/latex]
Cross-multiplying, [latex] 4x = 40 [/latex]
Simplifying, [latex] \displaystyle{x = \frac{40}{4}} [/latex]
Therefore, [latex] x = 10 [/latex]
| 1st Term | 2nd Term |
|---|---|
| 4 | 5 |
| 8 | x |
[latex] 6 : x = 10 : 25 [/latex]
Using fractional notation, [latex] \displaystyle{\frac{6}{x} = \frac{10}{25}} [/latex] or [latex] \displaystyle{\frac{6}{10} = \frac{x}{25}} [/latex]
Cross-multiplying, [latex] 150 = 10x [/latex]
Simplifying, [latex] \displaystyle{x = \frac{150}{10}} [/latex]
Therefore, [latex] x = 15 [/latex]
| 1st Term | 2nd Term |
|---|---|
| 6 | x |
| 10 | 25 |
[latex] x : 1.9 = 2.6 : 9.88 [/latex]
Using fractional notation, [latex] \displaystyle{\frac{x}{1.9} = \frac{2.6}{9.88}} [/latex] or [latex] \displaystyle{\frac{x}{2.6} = \frac{1.9}{9.88}} [/latex]
Cross-multiplying, [latex] 9.88x = 4.94 [/latex]
Simplifying, [latex] \displaystyle{x = \frac{4.94}{9.88}} [/latex]
Therefore, [latex] x = 0.5 [/latex]
| 1st Term | 2nd Term |
|---|---|
| x | 1.9 |
| 2.6 | 9.88 |
[latex] \displaystyle{3 : 3\frac{3}{4} = x : 5\frac{1}{4}} [/latex]
Rewriting as an improper fraction, [latex] \displaystyle{3 : \frac{15}{4} = x : \frac{21}{4}} [/latex]
Multiplying both sides by 4, [latex] 12 : 15 = 4x : 21 [/latex]
Using fractional notation, [latex] \displaystyle{\frac{12}{15} = \frac{4x}{21}} [/latex] or [latex] \displaystyle{\frac{12}{4x} = \frac{15}{21}} [/latex]
Cross-multiplying, [latex] 252 = 60x [/latex]
Simplifying, [latex] \displaystyle{x = \frac{252}{60}} [/latex]
Therefore, [latex] \displaystyle{x = \frac{21}{5} = 4\frac{1}{5}} [/latex]
| 1st Term | 2nd Term |
|---|---|
| 12 | 15 |
| 4x | 21 |
Express each of the following ratios in its simplest form:
The distance (in km) that Ben can walk in [latex] \displaystyle{3\frac{1}{2}} [/latex] hours.
How long (in hours and minutes) will it take him to walk 15 km?
Calculating the distance (in km) he can walk in [latex] \displaystyle{3\frac{1}{2}} [/latex] hours.
km : h = km : h
[latex] \displaystyle{9 : 2 = x : 3\frac{1}{2}} [/latex]
Rewriting as an improper fraction, [latex] \displaystyle{9 : 2 = x : \frac{7}{2}} [/latex]
Multiplying both sides by 2, [latex] 18 : 4 = 2x : 7 [/latex]
Using fractional notation, [latex] \displaystyle{\frac{18}{4} = \frac{2x}{7}} [/latex] or [latex] \displaystyle{\frac{18}{2x} = \frac{4}{7}} [/latex]
Cross-multiplying, [latex] 126 = 8x [/latex]
Simplifying, [latex] \displaystyle{x = \frac{126}{8}} [/latex]
[latex] x = 15.75 km [/latex]
Therefore, Ben can walk a distance of 15.75 km in [latex] \displaystyle{3\frac{1}{2}} [/latex] hours.
| 1st Term | 2nd Term |
|---|---|
| 18 | 4 |
| 2x | 7 |
Calculating the time (in hours and minutes) that it will take him to walk 15 km:
km : h = km : h
[latex] 9 : 2 = 15 : x [/latex]
Using fractional notation, [latex] \displaystyle{\frac{9}{2} = \frac{15}{x}} [/latex] or [latex] \displaystyle{\frac{9}{15} = \frac{2}{x}} [/latex]
Cross-multiplying, [latex] 9x = 30 [/latex]
Simplifying, [latex] \displaystyle{x = \frac{30}{9} = 3\frac{1}{3}} [/latex]
[latex] \displaystyle{x = 3 h + \left(\frac{1}{3} \times 60\right) min} [/latex]
[latex] x = 3 h 20 min [/latex]
Therefore, Ben can walk 15 km in 3 hours and 20 minutes.
| 1st Term | 2nd Term |
|---|---|
| 9 | 2 |
| 15 | x |
Andrew (A), Brandon (B), and Chris (C) decide to form a partnership to start a snow removal business together. A invests $31,500, B invests $42,000, and C invests $73,500. They agree to share the profits in the same ratio as their investments.
What is the ratio of their investments, expressed in lowest terms?
In the first year of running the business, A's profit was $27,000. What were B's and C's profits?
In the second year, their total profit was $70,000. How much would each of them receive from this total profit?
Ratio of their investments:
A : B : C
31,500 : 42,000 : 73,500 Dividing each term by the common factor of 100,
315 : 420 : 735 Dividing each term by the common factor of 5,
63 : 84 : 147 Dividing each term by the common factor of 7,
9 : 12 : 21 Dividing each term by the common factor of 3,
3 : 4 : 7
Therefore, the ratio of their investments is 3 : 4 : 7.
A's profit was $27,000. B's and C's profits are calculated using one of the two methods, as follows:
Method 1:
Ratio of Investment = Ratio of Profit
A : B : C = A : B : C
Substituting terms, [latex] 3 : 4 : 7 = 27,000 : x : y [/latex]
Using fractional notation, [latex] \displaystyle{\frac{3}{4} = \frac{27,000}{x}} [/latex] and [latex] \displaystyle{\frac{3}{7} = \frac{27,000}{y}} [/latex]
Cross-multiplying, [latex] 3x = 108,000 [/latex] [latex] 3y = 189,000 [/latex]
Simplifying, [latex] x = \$36,000.00 [/latex] [latex] y = \$63,000.00 [/latex]
Method 2:
Ratio of Investment = Ratio of Profit
A : B : C = A : B : C
Substituting terms, [latex] 3 : 4 : 7 = 27,000 : x : y [/latex]
Using fractional notation, [latex] \displaystyle{\frac{3}{27,000} = \frac{4}{x} = \frac{7}{y}} [/latex]
Hence, [latex] \displaystyle{\frac{3}{27,000} = \frac{4}{x}} [/latex] and [latex] \displaystyle{\frac{3}{27,000} = \frac{7}{y}} [/latex]
Cross-multiplying, [latex] 3x = 108,000 [/latex] [latex] 3y = 189,000 [/latex]
Simplifying, [latex] x = \$36,000.00 [/latex] [latex] y = \$63,000.00 [/latex]
Therefore, B's profit is $36,000 and C's profit is $63,000.
| 1st Term | 2nd Term | 3rd Term |
|---|---|---|
| 3 | 4 | 7 |
| 27,000 | x | y |
In the second year, their total profit was $70,000. The profit that each of them would receive is calculated by using one of the methods, as follows:
Method 1:
Since A, B, and C agreed to share profits in the same ratio as their investments, the ratio of their individual investments to their individual profits should be equal to the ratio of the total investment to the total profit.
By adding the ratio of their investments (3 + 4 + 7), we know that the total profit of $70,000 should be distributed over 14 units. Therefore,
Ratio of Investment = Ratio of Profit
A : B : C : Total = A : B : C : Total
Substituting terms, [latex] 3 : 4 : 7 : 14 = A : B : C : 70,000 [/latex]
Using fractional notation, [latex] \displaystyle{\frac{3}{14} = \frac{A}{70,000}} [/latex], [latex] \displaystyle{\frac{4}{14} = \frac{B}{70,000}} [/latex], [latex] \displaystyle{\frac{7}{14} = \frac{C}{70,000}} [/latex]
Cross-multiplying, [latex] 14A = 210,000 [/latex], [latex] 14B = 280,000 [/latex], [latex] 14C = 490,000 [/latex]
Simplifying, [latex] A = /$15,000.00 [/latex], [latex] B = /$20,000.00 [/latex], [latex] C = /$35,000.00 [/latex]
Method 2:
Ratio of Investment = Ratio of Profit
A : B : C : Total = A : B : C : Total
Substituting terms, [latex] 3 : 4 : 7 : 14 = A : B : C : 70,000 [/latex]
Using fractional notation, [latex] \displaystyle{\frac{3}{A} = \frac{4}{B} = \frac{7}{C} = \frac{14}{70,000}} [/latex]
Hence, [latex] \displaystyle{\frac{3}{A} = \frac{14}{70,000}} [/latex] [latex] \displaystyle{\frac{4}{B} = \frac{14}{70,000}} [/latex] [latex] \displaystyle{\frac{7}{C} = \frac{14}{70,000}} [/latex]
Cross-multiplying, [latex] 14A = 210,000 [/latex], [latex] 14B = 280,000 [/latex], [latex] 14C = 490,000 [/latex]
Simplifying, [latex] A = /$15,000.00 [/latex], [latex] B = /$20,000.00 [/latex], [latex] C = /$35,000.00 [/latex]
| 1st Term | 2nd Term | 3rd Term | 4th Term |
|---|---|---|---|
| 3 | 4 | 7 | 14 |
| A | B | C | 70,000 |
Method 3:
Sharing Quantities Using Ratios:
A's share [latex] \displaystyle{= \frac{3}{14} \times 70,000.00 = \$15,000.00} [/latex]
B's share [latex] \displaystyle{= \frac{4}{14} \times 70,000.00 = \$20,000.00} [/latex]
C's share [latex] \displaystyle{= \frac{7}{14} \times 70,000.00 = \$35,000.00} [/latex]
Therefore, A, B, and C will receive profits of $15,000, $20,000, and $35,000, respectively.
Pro-ration is defined as sharing or allocating quantities, usually amounts of money, on a proportional basis.
Consider an example where Sarah paid $690 for a math course but decided to withdraw from the course after attending only half of it. As she attended only half the course, the college decided to refund half of her tuition fee, [latex] \displaystyle{= \frac{\$690}{2} = \$345} [/latex]. As the college calculated the refund amount proportional to the time she attended the course, we say that the college refunded her tuition fee on a pro-rata basis.
A few examples where pro-rated calculations are used are:
Calculate the pro-rated insurance premium for seven months if the annual premium paid for car insurance is $2,250.
Ratio of the premiums paid:
Premium ($) : Time (months) = Premium ($) : Time (months)
Substituting terms, [latex] 2,250 : 12 = x : 7 [/latex]
Using fractional notation, [latex] \displaystyle{\frac{2,250}{12} = \frac{x}{7}} [/latex] or [latex] \displaystyle{\frac{2,250}{x} = \frac{12}{7}} [/latex]
Cross-multiplying, [latex] 15,750 = 12x [/latex]
Simplifying, [latex] \displaystyle{x = \frac{15,750}{12}} [/latex]
[latex] x = \$1,312.50 [/latex]
Therefore, the pro-rated premium for seven months is $1,312.50.
| 1st Term | 2nd Term |
|---|---|
| 2,250 | x |
| 12 | 7 |
Johnson paid $350 for a two-year, weekly subscription of a health journal. After receiving 18 issues of the journal in his second year, he decided to cancel his subscription. What should be the amount of his pro-rated refund? Assume 1 year = 52 weeks.
Johnson paid for 104 issues (2 × 52) and received 70 issues (52 + 18); therefore, he should be refunded for 34 issues (104 - 70).
Issues (#) : Cost ($) = Issues (#) : Cost ($)
Substituting terms, [latex] 104 : 350 = 34 : x [/latex]
Using fractional notation, [latex] \displaystyle{\frac{104}{350} = \frac{34}{x}} [/latex] or [latex] \displaystyle{\frac{104}{34} = \frac{350}{x}} [/latex]
Cross-multiplying, [latex] 104x = 11,900 [/latex]
Simplifying, [latex] \displaystyle{x = \frac{11,900}{104}} [/latex]
[latex] x = 114.423076... = \$114.42 [/latex]
Therefore, his refund should be $114.42.
| 1st Term | 2nd Term |
|---|---|
| 104 | 350 |
| 34 | x |
For Problems 1 and 2, determine whether the pairs of ratios are in proportion or not.
a. 6 : 9 and 14 : 21
b. 5 : 15 and 2 : 8
c. 18 : 24 and 12 : 16
d. 12 : 60 and 6 : 24
a. 9 : 12 and 4 : 3
b. 10 : 30 and 8 : 24
c. 14 : 20 and 28 : 42
d. 15 : 12 and 24 : 30
For Problems 3 to 6, solve the proportion equations for the unknown value.
a. [latex] x : 4 = 27 : 36 [/latex]
b. [latex] 24 : x = 6 : 9 [/latex]
c. [latex] 5 : 9 = x : 3 [/latex]
d. [latex] 1 : 2 = 5 : x [/latex]
a. [latex] x : 8 = 6 : 24 [/latex]
b. [latex] 3 : x = 18 : 42 [/latex]
c. [latex] 15 : 5 = x : 15 [/latex]
d. [latex] 28 : 35 = 4 : x [/latex]
a. [latex] \displaystyle{x : 18\frac{1}{4} = 8 : 11\frac{3}{4}} [/latex]
b. [latex] \displaystyle{7\frac{1}{5} : x = 5\frac{3}{4} : 3\frac{2}{5}} [/latex]
c. [latex] \displaystyle{1 : 4\frac{1}{2} = x : 2\frac{3}{4}} [/latex]
d. [latex] \displaystyle{1\frac{1}{2} : 2\frac{1}{4} = 1\frac{3}{4} : x} [/latex]
a. [latex] x : 3.65 = 5.5 : 18.25 [/latex]
b. [latex] 2.2 : x = 13.2 : 2.5 [/latex]
c. [latex] 4.25 : 1.87 = x : 2.2 [/latex]
d. [latex] 2.4 : 1.5 = 7.2 : x [/latex]
A truck requires 96 litres of gas to cover 800 km. How many litres of gas will it require to cover 1,500 km? br
Based on Alvin's past experience, it would take his team five months to complete two projects. How long would his team take to complete eight similar projects?
Eric paid a property tax of $3,600 for his land that measures 330 square metres. Using the same tax rate, what would his neighbour's property tax be if the size of the house is 210 square metres and is taxed at the same rate?
The city of Brampton charges $1,750 in taxes per year for a 2,000 square metre farm. How much would Maple Farms have to pay in taxes if they had a 12,275 square metre farm in the same area?
On a map, 4 cm represents 5.0 km. If the distance between Town A and Town B on the map is 9.3 cm, how many kilometres apart are these towns?
On a house plan, 1.25 cm represents 3 metres. If the actual length of a room is 5.4 metres, how will this length be represented in the plan in cm?
Steve invested his savings in a GIC, mutual funds, and a fixed deposit in the ratio of 5 : 4 : 3, respectively. If he invested $10,900 in mutual funds, calculate his investments in the GIC and the fixed deposit.
The ratio of the distance from An's house to Mark, Jeff, and Justin's houses is 3 : 5.25 : 2, respectively. If the distance from Ann's house to Mark's is 9.50 km, calculate the distance from Ann's house to Justin's and Ann's house to Jeff's.
A, B, and C, started a business with investments in the ratio of 5 : 4 : 3, respectively. A invested $25,000, and all three of them agreed to share profits in the ratio of their investments.
Calculate B and C's investments.
If A's profit was $30,000 in the first year, calculate B and C's profits.
How much would each of them receive if, in the second year, the total profit was $135,000?
A, B, and C formed a partnership and invested in the ratio of 7 : 9 : 5, respectively. They agreed to share the profit in the ratio of their investments. A invested $350,000.
Calculate B and C's investments in the partnership.
In the first year, if A made $38,500 in profit from the partnership, how much did B and C make?
If the partnership made a profit of $126,000 in the second year, calculate each partner's share of the profit.
A, B and C invested $35,000, $42,000, and $28,000, respectively, to start an e-learning business. They realized that they required an additional $45,000 for operating the business. How much did each of them have to individually invest to maintain their original investment ratio?
Three wealthy business partners decided to invest $150,000, $375,000, and $225,000, respectively, to purchase an industrial plot on the outskirts of the city. They required an additional $90,000 to build an industrial shed on the land. How much did each of them have to individually invest to maintain their original investment ratio?
Chris, Diane, and David invested a total of $520,000 in the ratio of 3 : 4 : 6, respectively, to start a business. Two months later, each of them invested an additional $25,000 into the business. Calculate their new investment ratio after the additional investments.
Michael and his two sisters purchased an office for $720,000. Their individual investments in the office were in the ratio of 5 : 4 : 3, respectively. After the purchase, they decided to renovate the building and purchase furniture, so each of them invested an additional $60,000. Calculate their new investment ratio after the additional investments.
A student pays $620 for a course that has 25 classes. Calculate the pro-rated refund she would receive if she only attends 5 classes before withdrawing from the course.
Megan joined a driving school that charges $375 for 12 classes. After attending 7 classes, she decided that she did not like the training and wanted to cancel the remaining classes. Calculate the pro-rated refund she should receive.
Frank bought a brand new car on August 01, 2018 and obtained pre-paid insurance of $1,058 for the period of August 01, 2018 to July 31, 2019. After two months of using the car, he sold it and cancelled his insurance. Calculate the pro-rated refund he should receive from the insurance company.
The owner of a new gaming business decided to insure his servers and computers. His insurance company charged him a premium of $2,000 per quarter, where the first quarter starts on January 01. If his insurance is to start on February 01, how much pro-rated insurance premium did he have to pay for the rest of the first quarter? (Hint: A 'quarter' of a year is three months).
If the annual salary of an employee is $45,000, calculate his bi-weekly salary using pro-rations. Assume that there are 52 weeks in a year and 26 bi-weekly payments.
Ashley received a job offer at a company that would pay her $2,800, bi-weekly. What would her annual salary be, assuming that she would receive 26 payments in a year?
Charles set up a new charity fund to support children in need. For every $10 collected by the charity, the government donated an additional grant of $5 to the charity. At the end of three months, if his charity fund had a total of $135,000, including the government grant, calculate the amount the charity received from the government.
The tax on education materials sold in Ontario is such that for every $1.00 worth of materials sold, the buyer would have to pay an additional $0.05 in taxes. If $25,000 worth of textbooks were sold at a bookstore before taxes, calculate the total amount of tax to be paid by the purchasers.
A first semester class in a college has six more girls than boys, and the ratio of the number of girls to boys in the class is 8 : 5.
How many students are there in the class?
If four girls and three boys joined the class, determine the new ratio of girls to boys in the class.
The advisory board of a public sector company has ten more men than women, and the ratio of the number of men to women is 8 : 3.
How many people are there on the board?
If four men and four women joined the board, determine the new ratio of men to women.
To estimate the number of tigers in a forest, a team of researchers tagged 84 tigers and released them into the forest. Six months later, 30 tigers were spotted, out of which 7 had tags. How many tigers were estimated to be in the forest?
Researchers were conducting a study to estimate the number of frogs in a pond. They put a bright yellow band on the legs of 60 frogs and released them into the pond. A few days later, 15 frogs were spotted, out of which 5 had bands. How many frogs were estimated to be in the pond?
