LemonmathSelf-study sheet

Grade 9 · Lesson 1.3

Solving word problems with simultaneous equations

Name: Date: Time spent: min

Goals

Know the three steps for solving a word problem with simultaneous equations.
Choose the unknowns, state their conditions and write the other quantities in terms of them.
Solve problems about numbers, motion, percentages and mixing solutions.

Remember

Step 1: choose two unknowns, state their conditions, write the quantities in terms of them, form two equations.
Step 2: solve the equations (substitution or elimination).
Step 3: check against the conditions, then answer, with units.
$\overline{ab}=10a+b$ ; downstream $x+y$ , upstream $x-y$ ; an increase of $p\%$ means multiplying by $1+\dfrac{p}{100}$ .

Exercises

Exercise 1

There are $36$ chickens and rabbits, with $100$ legs altogether. How many chickens and how many rabbits are there?

Exercise 2

Class 9B visits a museum: $42$ people, teachers and students together. A student ticket costs $20$ thousand đồng, a teacher ticket costs $40$ thousand đồng, and the tickets cost $920$ thousand đồng in total. How many students and how many teachers are there?

Exercise 3

In a two-digit number, the tens digit is twice the units digit. If the two digits are swapped, the new number is $36$ less than the original number. Find the number.

Exercise 4

A rectangular garden has perimeter $56$ m. If its length is increased by $2$ m and its width is decreased by $1$ m, its area stays the same. Find the length and the width of the garden.

Exercise 5

How many grams of a $10\%$ salt solution and how many grams of a $30\%$ salt solution must be mixed to make $200$ g of a $16\%$ salt solution?

Exercise 6 Review

Solve the chickens-and-rabbits problem of exercise 1 again by forming one equation in one unknown (as in Grade 8), then compare with the method of simultaneous equations.

Exercise 7

Last month two teams sewed $1000$ shirts. This month team A sewed $10\%$ more and team B sewed $5\%$ fewer than last month, so together they sewed $1040$ shirts.

a) Let $x$ and $y$ be the numbers of shirts sewn last month by team A and team B. Form the simultaneous equations.

b) Solve them to find $x$ and $y$ .

c) Which team sewed more this month, and by how many shirts?

Exercise 8 Challenge

Hà cycles from home to school. At $15$ km/h she arrives $6$ minutes before the start of lessons; at $12$ km/h she arrives $3$ minutes late. Find the distance from home to school and the time Hà needs to arrive exactly on time.

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. Two numbers have sum

50

and difference

14

. The smaller number is

18

.B.

32

.C.

36

.D.

25

Question 2. A rectangular garden has perimeter

40

m, and its length is

4

m more than its width. Let the length be

x

and the width

y

(m). The correct equations are

\begin{cases} x+y=40\\ x-y=4 \end{cases}

\begin{cases} x+y=20\\ x-y=4 \end{cases}

\begin{cases} 2x+y=40\\ x-y=4 \end{cases}

\begin{cases} x+y=20\\ y-x=4 \end{cases}

Question 3. A motor boat goes at

30

km/h downstream and

22

km/h upstream. The speed of the current is

4

km/h.B.

8

km/h.C.

26

km/h.D.

2

km/h.

Question 4. Two items have list prices of

x

and

y

thousand đồng. The first is reduced by

10\%

and the second by

20\%

, and the total to pay is

170

thousand đồng. The correct equation is

0.9x+0.8y=170

.B.

0.1x+0.2y=170

.C.

(x-10)+(y-20)=170

.D.

0.9x+0.2y=170

Part II. True or false

Question 1. There are

30

chickens and rabbits with

84

legs altogether. Let

x

be the number of chickens and

y

the number of rabbits.

a) $x+y=30$ .	T / F
b) $4x+2y=84$ .	T / F
c) There are $12$ rabbits.	T / F
d) There are $6$ more chickens than rabbits.	T / F

Part III. Short answer

Question 1. A two-digit number has digits that add up to

9

. Swapping the digits gives a number

45

more than the original. Find the number.

Question 2. A

10\%

salt solution is mixed with a

25\%

salt solution to make

300

g of a

20\%

solution. How many grams of the

10\%

solution are needed?

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: a chicken has $2$ legs and a rabbit has $4$ .

Steps:

Let $x$ be the number of chickens and $y$ the number of rabbits.
Form the equations from the number of animals and the number of legs, then solve.

Exercise 2.

What you need: the number of people and the cost give two equations.

Steps:

Let $x$ be the number of students and $y$ the number of teachers.
Form the equations, divide the cost equation by $20$ , then solve.

Exercise 3.

What you need: $\overline{ab}=10a+b$ ; $(10a+b)-(10b+a)=9(a-b)$ .

Steps:

Form $a=2b$ and $9(a-b)=36$ .
Substitute $a=2b$ into the second equation.

Exercise 4.

What you need: perimeter $2(x+y)$ ; area $xy$ . When you expand $(x+2)(y-1)=xy$ , the $xy$ terms cancel.

Steps:

Let the length be $x$ and the width $y$ (m, $x\ge y>1$ ).
Form $x+y=28$ and $(x+2)(y-1)=xy$ , and simplify the second equation.
Solve the equations.

Exercise 5.

What you need: mass of salt $=$ mass of solution $\times$ concentration; the total mass of solution and the total mass of salt do not change when you mix.

Steps:

Let $x$ and $y$ be the grams of the $10\%$ and the $30\%$ solutions.
Form the equations from the total mass and the total salt ( $16\%$ of $200$ g is $32$ g).

Exercise 6.

What you need: if there are $x$ chickens, there are $36-x$ rabbits.

Steps:

Write the number of legs in terms of $x$ .
Solve the linear equation in one unknown.

Exercise 7.

What you need: $10\%$ more means multiplying by $1.1$ ; $5\%$ fewer means multiplying by $0.95$ .

Steps:

Form $x+y=1000$ and $1.1x+0.95y=1040$ .
Multiply the second equation by $20$ , then substitute $x=1000-y$ .
Work out each team's shirts this month.

Exercise 8.

What you need: distance $=$ speed $\times$ time; in hours, $6$ minutes $=0.1$ h and $3$ minutes $=0.05$ h.

Steps:

Let $s$ be the distance (km) and $t$ the time to arrive exactly on time (hours), $t>0.1$ .
Form $s=15(t-0.1)$ and $s=12(t+0.05)$ .
Solve by substitution.

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