randRange( 0, 2 ) [randRange( 2, 4 ), randRange( 2, 4 ), randRange( 40, 50 )][INDEX] [randRange( X + 2, 10 ), randRange( X + 2, 10 ), randRange( 30, 50 )][INDEX] randRange( 0, 1 ) randRange( 0, 1 ) [roundTo( 8, randRange( 5, 8 ) + CENT_1 * 0.5 ), roundTo( 8, randRange( 5, 8 ) + CENT_1 * 0.5 ), 1][INDEX] [roundTo( 8, randRange( 2, 4 ) + CENT_2 * 0.5 ), roundTo( 8, randRange( 2, 4 ) + CENT_2 * 0.5 ), 1][INDEX] roundTo( 8, A1 * X + B1 * Y ) 1 [-1, -1, randRange( -4, -2 )][INDEX] roundTo( 8, A2 * X + B2 * Y ) A1 * C2 > 0 ? "-" : "+" B1 * Y > 0 ? "-" : "+" [ $._("A group of adults and kids went to see a movie"), $._("A few families took a trip to an amusement park together"), "" ][INDEX] [ $._("Find the number of adults and kids in the group"), $._("Find the number of adults and kids on the trip"), "" ][INDEX] [ $._("adults"), $._("adults"), $._("home team fans") ][INDEX] [ $._("kids"), $._("kids"), $._("away team fans") ][INDEX]

PROBLEM_1. Tickets cost `$localeToFixed(A1, 2)` each for adults and `$localeToFixed(B1, 2)` each for kids, and the group paid `$localeToFixed(C1, 2)` in total. There were `abs(C2)` fewer adults than kids in the group.

PROBLEM_2.

`C1` people attended a baseball game. Everyone there was a fan of either the home team or the away team. The number of home team fans was `abs(C2)` less than `abs(B2)` times the number of away team fans.

How many home team and away team fans attended the game?

# of UNIT_1 = X

# of UNIT_2 = Y

Let `x` equal the number of UNIT_1 and `y` equal the number of UNIT_2.

The system of equations is then:

`\blue{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}`
`\green{x = expr(["+", ["*", -B2, "y"], C2])}`

Solve for `x` and `y` using substitution.

Since `x` has already been solved for, substitute `\green{expr(["+", ["*", -B2, "y"], C2])}` for `x` in the first equation.

`\blue{A1-}\green{(expr(["+", ["*", -B2, "y"], C2]))}\blue{+ expr(["*", B1, "y"]) = C1}`

Simplify and solve for `y`.

`expr(["+", ["*", roundTo(8, A1 * -B2), "y"], roundTo(8, A1 * C2)]) + expr(["*", B1, "y"]) = C1`

`expr(["+", ["*", roundTo(8, A1 * -B2 + B1), "y"], roundTo(8, A1 * C2)]) = C1`

`expr(["*", roundTo(8, A1 * -B2 + B1), "y"]) = roundTo(8, C1 - A1 * C2)`

`y = \dfrac{roundTo(8, C1 - A1 * C2)}{roundTo( 8, A1 * -B2 + B1 )}`

`\orange{y = Y}`

Now that you know `\orange{y = Y}`, plug it back into `\green{x = expr(["+", ["*", -B2, "y"], C2])}` to find `x`.

`\green{x = -B2-}\orange{(Y)}\green{ + C2}`

`x = -B2 * Y + C2`

`\red{x = X}`

You can also plug `\orange{y = Y}` into `\blue{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}` and get the same answer for `x`:

`\blue{expr(["*", A1, "x"]) + B1-}\orange{(Y)}\blue{= C1}`

`\red{x = X}`

There were `X` UNIT_1 and `Y` UNIT_2.

randRange( 0, 2 ) [randRange( 2, 4 ), randRange( 2, 4 ), randRange( 40, 50 )][INDEX] [randRange( X + 2, 10 ), randRange( X + 2, 10 ), randRange( 30, 50 )][INDEX] randRange( 0, 1 ) randRange( 0, 1 ) [roundTo( 8, randRange( 5, 8 ) + CENT_1 * 0.5 ), roundTo( 8, randRange( 5, 8 ) + CENT_1 * 0.5 ), 1][INDEX] [roundTo( 8, randRange( 2, 4 ) + CENT_2 * 0.5 ), roundTo( 8, randRange( 2, 4 ) + CENT_2 * 0.5 ), 1][INDEX] roundTo( 8, A1 * X + B1 * Y ) 1 [-1, -1, randRange( -4, -2 )][INDEX] roundTo( 8, A2 * X + B2 * Y ) B1 * C2 > 0 ? "-" : "+" A1 * X > 0 ? "-" : "+" [ $._("The drama club sold bags of candy and cookies to raise money for the spring show"), $._("A local gift shop sold bags of candy and cookies for Halloween"), "" ][INDEX] [ $._("Find the number of bags of candy and cookies sold by the drama club"), $._("Find the number of bags of candy and cookies sold by the gift shop"), "" ][INDEX] [ $._("bags of candy"), $._("bags of candy"), $._("angle 1") ][INDEX] [ $._("bags of cookies"), $._("bags of cookies"), $._("angle 2") ][INDEX]

PROBLEM_1. Bags of candy cost `$localeToFixed(A1, 2)`, and bags of cookies cost `$localeToFixed(B1, 2)`, and sales equaled `$localeToFixed(C1, 2)` in total. There were `C2` more bags of cookies than candy sold.

PROBLEM_2.

The sum of two angles is `C1^\circ`. Angle 2 is `abs(C2)^\circ` smaller than `abs(A2)` times angle 1.

What are the measures of the two angles in degrees?

# of UNIT_1 = X

# of UNIT_2 = Y

Let `x` equal the number of UNIT_1 and `y` equal the number of UNIT_2.

Let `x` equal the measure of UNIT_1 and `y` equal the measure of UNIT_2.

The system of equations is then:

`\blue{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}`
`\green{y = expr(["+", ["*", -A2, "x"], C2])}`

Since we already have solved for `y` in terms of `x`, we can use substitution to solve for `x` and `y`.

Substitute `\green{expr(["+", ["*", -A2, "x"], C2])}` for `y` in the first equation.

`\blue{expr(["*", A1, "x"]) + B1-}\green{(expr(["+", ["*", -A2, "x"], C2]))}\blue{= C1}`

Simplify and solve for `x`.

`expr(["+", ["*", A1, "x"], ["*", roundTo(8, B1 * -A2), "x"]]) + roundTo(8, B1 * C2) = C1`

`expr(["+", ["*", roundTo(8, A1 + B1 * -A2), "x"], roundTo(8, B1 * C2)]) = C1`

`expr(["*", roundTo(8, A1 + B1 * -A2), "x"]) = roundTo(8, C1 - B1 * C2)`

`x = \dfrac{roundTo( 8, C1 - B1 * C2 )}{roundTo(8, A1 + B1 * -A2)}`

`\red{x = X}`

Now that you know `\red{x = X}`, plug it back into `\green{y = expr(["+", ["*", -A2, "x"], C2])}` to find `y`.

`\green{y = -A2-}\red{(X)}\green{ + C2}`

`y = roundTo(8, -A2 * X) + C2`

`\orange{y = Y}`

You can also plug `\red{x = X}` into `\blue{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}` and get the same answer for `y`:

`\blue{A1-}\red{(X)}\blue{ + expr(["*", B1, "y"]) = C1}`

`\orange{y = Y}`

`X` bags of candy and `Y` bags of cookies were sold.

The measure of angle 1 is `X^\circ` and the measure of angle 2 is `Y^\circ`.

randRange( 2, 4 ) randRange( X + 2, 10 ) 1 randRange( 0, 1 ) randRange( 0, 1 ) roundTo( 8, randRange( 100, 300 ) / 100 ) INDEX === 0 ? -1 * Z1 : randRangeWeighted( -4, 4, -1, 0 ) * Z1 roundTo( 8, randRange( 200, 600 ) / 100 ) roundTo( 8, randRange( 200, 600 ) / 100 ) roundTo( 8, randRange( 200, 600 ) / 100 ) roundTo( 8, randRange( 200, 600 ) / 100 ) roundTo( 8, randRange( 5, 8 ) + CENT_1 * 0.5 ) roundTo( 8, randRange( 2, 4 ) + CENT_2 * 0.5 ) roundTo( 8, A1 * X + B1 * Y ) A1 * randRange( 2, 4 ) roundTo( 8, randRange( 7, 12 ) + ( CENT_1 + CENT_2 ) * 0.5 ) roundTo( 8, A2 * X + B2 * Y ) [ 1, roundTo( 8, -A2 / A1 ), 1, A2 > 0 && A1 < 0 ? A2 : -A2 ][ INDEX ] [ 1, 1, roundTo( 8, -A1 / A2 ), A2 > 0 && A1 < 0 ? -A1 : A1 ][ INDEX ] B1 * Y > 0 ? "-" : "+" B2 * Y > 0 ? "-" : "+"

All of the {3rd|4th|5th} grade teachers and students from school(1) went on a field trip to an {art|archaeology} museum. Tickets were `$localeToFixed(A1, 2)` each for teachers and `$localeToFixed(B1, 2)` each for students, and the group paid `$localeToFixed(C1, 2)` in total.

{A few weeks later|The next month}, the same group visited a {science|natural history} museum where the tickets cost `$localeToFixed(A2, 2)` each for teachers and `$localeToFixed(B2, 2)` each for students, and the group paid `$localeToFixed(C2, 2)` in total.

Find the number of teachers and students on the field trips.

# of teachers = X

# of students = Y

Let `x` equal the number of teachers and `y` equal the number of students.

The system of equations is:

`\blue{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}`
`\green{expr(["+", ["*", A2, "x"], ["*", B2, "y"]]) = C2}`

Solve for `x` and `y` using elimination.

Multiply the bottom equation by `MULT2`.

Multiply the top equation by `MULT1` and the bottom equation by `MULT2`.

Multiply the top equation by `MULT1`.

`\blue{expr(["+", ["*", A1 * MULT1, "x"], ["*", B1 * MULT1, "y"]]) = C1 * MULT1}`
`\green{expr(["+", ["*", A2 * MULT2, "x"], ["*", B2 * MULT2, "y"]]) = C2 * MULT2}`

Add the top and bottom equations together.

`expr(["*", roundTo(8, B1 * MULT1 + B2 * MULT2), "y"]) = roundTo(8, C1 * MULT1 + C2 * MULT2)`

`y = \dfrac{roundTo(8, C1 * MULT1 + C2 * MULT2)}{roundTo(8, B1 * MULT1 + B2 * MULT2)}`

`\orange{y = Y}`

Now that you know `\orange{y = Y}`, plug it back into `\blue{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}` to find `x`.

`\blue{expr(["*", A1, "x"]) + B1-}\orange{(Y)}\blue{= C1}`

`expr(["+", ["*", A1, "x"], B1 * Y]) = C1`

`expr(["*", A1, "x"]) = roundTo(8, C1 - B1 * Y)`

`x = \dfrac{roundTo( 8, C1 - B1 * Y )}{A1}`

`\red{x = X}`

You can also plug `\orange{y = Y}` into `\green{expr(["+", ["*", A2, "x"], ["*", B2, "y"]]) = C2}` and get the same answer for `x`:

`\green{expr(["*", A2, "x"]) + B2-}\orange{(Y)}\green{= C2}`

`\red{x = X}`

There were `X` teachers and `Y` students on the field trips.

randRange( 11, 99 ) randRange( 10, X - 1 ) 0 roundTo( 8, randRange( 100, 300 ) / 100 ) INDEX === 0 ? -1 * Z1 : randRangeWeighted( -4, 4, -1, 0 ) * Z1 roundTo( 8, randRange( 200, 600 ) / 100 ) roundTo( 8, randRange( 200, 600 ) / 100 ) roundTo( 8, randRange( 200, 600 ) / 100 ) roundTo( 8, randRange( 200, 600 ) / 100 ) 1 [ 1, Z1, Z2, Z5 ][ INDEX ] roundTo( 8, A1 * X + B1 * Y ) 1 [ -1, Z2, Z1, Z6 ][ INDEX ] roundTo( 8, A2 * X + B2 * Y ) [ 1, roundTo( 8, -B2 / B1 ), 1, B2 > 0 && B1 < 0 ? B2 : -B2 ][ INDEX ] [ 1, 1, roundTo( 8, -B1 / B2 ), B2 > 0 && B1 < 0 ? -B1 : B1 ][ INDEX ] A1 * X > 0 ? "-" : "+" A2 * X > 0 ? "-" : "+" max(X, Y) min(X, Y)

The sum of two numbers is `C1`, and their difference is `C2`. What are the two numbers?

larger number = LARGER

smaller number = SMALLER

Let `x` be the first number, and let `y` be the second number.

The system of equations is:

`\blue{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}`
`\green{expr(["+", ["*", A2, "x"], ["*", B2, "y"]]) = C2}`

Solve for `x` and `y` using elimination.

Multiply the bottom equation by `MULT2`.

Multiply the top equation by `MULT1` and the bottom equation by `MULT2`.

Multiply the top equation by `MULT1`.

`\blue{expr(["+", ["*", A1 * MULT1, "x"], ["*", B1 * MULT1, "y"]]) = C1 * MULT1}`
`\green{expr(["+", ["*", A2 * MULT2, "x"], ["*", B2 * MULT2, "y"]]) = C2 * MULT2}`

Add the top and bottom equations together.

`expr(["*", roundTo(8, A1 * MULT1 + A2 * MULT2), "x"]) = roundTo(8, C1 * MULT1 + C2 * MULT2)`

`x = \dfrac{roundTo(8, C1 * MULT1 + C2 * MULT2)}{roundTo( 8, A1 * MULT1 + A2 * MULT2 )}`

`\red{x = X}`

Now that you know `\red{x = X}`, plug it back into `\blue{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}` to find `y`.

`\blue{A1-}\red{(X)}\blue{ + expr(["*", B1, "y"]) = C1}`

`expr(["+", A1 * X, ["*", B1, "y"]]) = C1`

`expr(["*", B1, "y"]) = roundTo( 8, C1 - A1 * X )`

`\dfrac{expr(["*", B1, "y"])}{\blue{B1}} = \dfrac{roundTo( 8, C1 - A1 * X )}{\blue{B1}}`

`\orange{y = Y}`

You can also plug `\red{x = X}` into `\green{expr(["+", ["*", A2, "x"], ["*", B2, "y"]]) = C2}` and get the same answer for `y`:

`\green{A2-}\red{(X)}\green{ + expr(["*", B2, "y"]) = C2}`

`\orange{y = Y}`

Therefore, the larger number is `LARGER`, and the smaller number is `SMALLER`.