Solve word problems using equations

{"questions":[{"content":"One team harvested $8\\space ha,$ of land, and another harvested $9\\space ha.$ The second team's yield per hectare was $2 \\space t$ less than the first team's. What was the first team's yield per hectare if they harvested $136\\space t$ altogether?[[image-1]][[fill_choice_big-511]]","widgets":{"image-1":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/10/ovoshhi-prazdnik-urozhai-tykva-baklozhan-tomat-perets-luk-3.svg","width":"299"},"fill_choice_big-511":{"type":"fill_choice_big","options":["$8\\space t$","$12\\space t$","$20\\space t$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the first team collect $x\\space t$ crops from each $ha.$<br />Then the second team collected $x-2\\space t.$","The first team harvested from $8 \\space ha$: $8x.$<br />The second team harvested from $9 \\space ha$: $9(x-2)$<br />Both teams together collected $136\\space t$ harvest: $$8x+9(x-2)=136$$","Let's solve the resulting equation: $$8x+9(x-2)=136$$ $$8x+9x-18=136$$ $$17x=136$$ $$x=136:17=8$$"]},{"content":"A boat leaves point $A$ at $25\\space km/h.$ After $3\\space h$, a second boat leaves point $B$ and travels towards the first at $30\\space km/h.$ How long after the first boat's departure will they meet if the points are $185\\space km?$ apart?[[image-4]][[fill_choice_big-542]]","widgets":{"image-4":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/03/yahta-lodka-korabl.svg","width":"299"},"fill_choice_big-542":{"type":"fill_choice_big","options":["$5$","$3$","$4$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the time of movement of the first steamer as $x.$<br />Then the second ship was on the move three hours less: $x-3$","Together, the boats travel $185\\space km$ before meeting: $$25x+30(x-3)=185$$","Let's solve the resulting equation: $$25x+30(x-3)=185$$ $$25x+30x-90=185$$ $$55x=275$$ $$x=275:55=5$$"]},{"content":"To prepare the pie, we bought $4 \\space kg$ bananas and $5\\space kg$ apples. The price of a kilogram of apples is $10\\space \\text{rubles}.$ lower than the price of a kilogram of bananas. At what price were bananas bought if the total price paid for the purchase was $1030\\space \\text{rubles}.?$[[image-9]][[fill_choice_big-593]]","widgets":{"image-9":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/03/pirog-yabloko-yablochnyj-pirog.svg","width":"300"},"fill_choice_big-593":{"type":"fill_choice_big","options":["$120 \\space \\text{rubles}.$","$100\\space \\text{rubles}.$","$60\\space \\text{rubles}.$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the price of bananas as $x,$ then apples cost $x-10.$","To prepare the pie, we bought $4 \\space kg$ bananas and $5\\space kg$ apples and paid $1030\\space \\text{rubles}.$:$$4x+5(x-10)=1030$$","Let's solve the resulting equation: $$4x+5(x-10)=1030$$ $$4x+5x-50=1030$$ $$9x=1080$$ $$x=1080:9=120$$"]},{"content":"There are $125$ books on three shelves. The second shelf has $2$ times as many books as the first, and the third has $7$ fewer books than the first. How many books are on the first shelf?[[image-16]][[fill_choice_big-634]]","widgets":{"image-16":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/12/knizhnyj-shkaf.svg","width":"299"},"fill_choice_big-634":{"type":"fill_choice_big","options":["$33$","$24$","$38$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let there be $x$ books on the first shelf.<br />Then on the second shelf there are $2x$ books.<br />The third shelf has $7$ fewer books than the first: $x-7.$<br />","In total there are $125$ books on the shelves: $$x+2x+(x-7)=125$$","Let's solve the resulting equation: $$x+2x+(x-7)=125$$ $$3x+x-7=125$$ $$4x=132$$ $$x=132:4=33$$"]},{"content":"A truck travels the same distance in $6\\space h$ as a car does in $5 \\space h.$ Find the truck's speed if it is $21\\space km/h.$ slower than the car.[[image-100]][[fill_choice_big-675]]","widgets":{"image-100":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/10/gruzovik-samosval-mashina-1.svg","width":"299"},"fill_choice_big-675":{"type":"fill_choice_big","options":["$105\\space km/h$","$86\\space km/h$","$63\\space km/h$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the speed of the truck as $x.$<br />Then the speed of the passenger car will be $x+21.$","A truck travels the same distance in $6\\space h$ as a car does in $5 \\space h$: $$6x=5(x+21)$$","Let's solve the resulting equation: $$6x=5(x+21)$$ $$6x=5x+105$$ $$x=105$$"]},{"content":"The first train has $4$ times as many carriages as the second. $27$ carriages are moved from the first train to the second, giving both trains the same number. How many carriages were originally on the first train?[[image-81]][[fill_choice_big-716]]","widgets":{"image-81":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/06/station.svg","width":"298"},"fill_choice_big-716":{"type":"fill_choice_big","options":["$72$","$27$","$38$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the number of cars in the second train as $x.$<br />The first train had $4$ times as many carriages: $4x$","To make the number of cars on the trains equal, $27$ cars were unhooked from the first train and attached to the second: $$4x-27=x+27$$","Let's solve the resulting equation: $$4x-27=x+27$$ $$4x-x=27+27$$ $$3x=54$$ $$x=54:3=18$$","The first train had $4x$ cars: $$18\\cdot 4 = 72$$"]},{"content":"The first ticket office sells $58$ hockey tickets, and $32$ extra tickets are delivered to the second office. Both offices then have the same number of tickets. How many tickets were originally in the second office if the first initially had $2$ times as many?[[image-64]][[fill_choice_big-757]]","widgets":{"image-64":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/06/shop-assistant-1.svg","width":"300"},"fill_choice_big-757":{"type":"fill_choice_big","options":["$90$","$45$","$50$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the number of tickets at the second box office as $x.$<br />Then at the first box office there were $2x$ tickets.","The first office sells $58$ tickets and the second receives $32$ more tickets. Both offices then have the same number: $$2x-58=x+32$$","Let's solve the resulting equation: $$2x-58=x+32$$ $$2x-x=32+58$$ $$x=90$$"]},{"content":"The trip from home to the dacha lasted $2 \\space h \\space30\\space min.$ If the speed of the car was greater by $20\\space km/h,$ then the trip would take $2\\space h.$ What is the speed of the car? [[image-49]][[fill_choice_big-798]]","widgets":{"image-49":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/04/zapravka-1.svg","width":"299"},"fill_choice_big-798":{"type":"fill_choice_big","options":["$80\\space km/h$","$60\\space km/h$","$65\\space km/h$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's convert $30$ minutes into hours: $$\\frac{30}{60}=0.5$$Then $2 \\space h \\space30\\space min = 2.5\\space h$","Let the car's speed be $x.$ In $2.5\\space h$ it travels $2.5x \\space km.$<br />If the speed of the car was higher by $20\\space km/h,$ then the trip would take $2\\space h$:$$(x+20)\\cdot 2 = 2.5x$$","Let's solve the resulting equation: $$2x+40 = 2.5x$$ $$2.5x-2x=40$$ $$0.5x=40$$ $$x=40:0.5=80$$"]},{"content":"The second bus initially has $2$ times as many passengers as the first. At a stop, $7$ passengers leave the first bus and $16$ leave the second. Both buses then have the same number of passengers. How many passengers were originally on the first bus?[[image-36]][[fill_choice_big-839]]","widgets":{"image-36":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/11/obrazavr_school-bus.svg","width":"300"},"fill_choice_big-839":{"type":"fill_choice_big","options":["$9$","$11$","$6$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the number of passengers on the first bus as $x.$<br />Then there were $2x$ passengers on the second bus.","When at the stop $7$ people got off the first bus, and $16$ people got off the second, then there were equal numbers of people on both buses: $$x-7=2x-16$$","Let's solve the resulting equation: $$x-7=2x-16$$ $$2x-x=16-7$$ $$x=9$$"]},{"content":"A roll of fabric $70 \\space m$ long is cut into two pieces. Find the length of the first piece if $0.4$ of its length equals $0.3$ of the second piece's length.[[image-25]][[fill_choice_big-880]]","widgets":{"image-25":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/03/arhitektor-stroitel-doska-chertezh-rulon-bumaga-3.svg","width":"300"},"fill_choice_big-880":{"type":"fill_choice_big","options":["$30 \\space m$","$11 \\space m$","$28 \\space m$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the length of the first piece as $x.$<br />Then the length of the second piece will be $70-x.$","$0.4$ of the first piece's length equals $0.3$ of the second piece's length: $$0.4x=0.3(70-x)$$","Let's solve the resulting equation: $$0.4x=0.3(70-x)$$ $$0.4x=21-0.3x$$ $$0.4x+0.3x=21$$ $$0.7x=21$$ $$x=21:0.7=30$$"]}]}

{"questions":[{"content":"The first warehouse contains $272 \\space t$ of coal and the second contains $580 \\space t.$ Each day, the first ships out $12\\space t$ of coal and the second ships out $30\\space t.$ After how many days will the second warehouse have $2$ times as much coal as the first?[[image-1]][[fill_choice_big-511]]","widgets":{"image-1":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2021/08/Obrazavr-s-lopatoj.svg","width":"300"},"fill_choice_big-511":{"type":"fill_choice_big","options":["$6$","$5$","$4$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the number of days as $x.$<br />","The first warehouse ships out $12\\space t$ of coal per day and the second ships out $30\\space t,$ The first must end up with half as much coal as the second:$$2(272-12x)=580-30x$$","$$544-24x=580-30x$$ $$6x=36$$ $$x=36:6=6$$"]},{"content":"A triangle has a perimeter of $28\\space cm.$ One side is $3\\space cm$ shorter than another and $2$ times as long as the third. Find the shortest side.[[image-4]][[fill_choice_big-542]]","widgets":{"image-4":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/08/obrazavr-s-uglom-krivye.svg","width":"299"},"fill_choice_big-542":{"type":"fill_choice_big","options":["$5\\space cm$","$2 \\space cm$","$13\\space cm$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the smaller side as $x.$<br />Then the first side is twice as large: $2x.$<br />The second side is $3\\space cm$ larger than the first: $2x+3.$","Perimeter of triangle $28\\space cm$:$$x+2x+(2x+3)=28$$","Let's solve the resulting equation: $$x+2x+2x+3=28$$ $$5x=25$$ $$x=25:5=5$$"]},{"content":"A company rents three rooms with a total area of $162\\space m^2.$ One room has $2$ times the area of another and an area $12\\space m^2$ smaller than the third. Find the area of the smallest room.[[image-9]][[fill_choice_big-593]]","widgets":{"image-9":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/03/arhitektor-stroitel-doska-1-1.svg","width":"300"},"fill_choice_big-593":{"type":"fill_choice_big","options":["$30$","$120$","$20$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the area of the smaller room as $x.$<br />Then the first room will be twice as large: $2x.$<br />The second room is $12 \\space m^2$ larger than the area of the first: $2x+12.$","Total premises occupy the area of $162\\space m^2$: $$x+2x+(2x+12)=162$$","$$x+2x+2x+12=162$$ $$5x=150$$ $$x=150:5=30$$"]},{"content":"By mowing $40\\space ha$ daily instead of $30\\space ha,$, the team was able to mow the meadow one day faster than planned. What is the area of the meadow?[[image-100]][[fill_choice_big-675]]","widgets":{"image-100":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/01/park-1.svg","width":"299"},"fill_choice_big-675":{"type":"fill_choice_big","options":["$120\\space ha$","$110\\space ha$","$280\\space ha$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the number of days as $x. $<br />Then the crew had to mow $30x\\space ha.$","However, the team mows $40 \\space ha$ per day and finishes one day earlier: $$30x=40(x-1)$$","Let's solve the resulting equation: $$30x=40x-40$$ $$10x=40$$ $$x=40:10=4$$<br />In $4$ days the crew had to mow: $$4\\cdot 30=120$$"]},{"content":"Having increased the average speed from $200$ to $250\\space m/min,$, the athlete began to run the distance $2\\space min$ faster. What is the length of the course?[[image-81]][[fill_choice_big-716]]","widgets":{"image-81":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/10/run.svg","width":"298"},"fill_choice_big-716":{"type":"fill_choice_big","options":["$2000 \\space m$","$3000 \\space m$","$4000 \\space m$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the time to complete the distance as $x.$<br />Then the length of the distance will be $200x.$","When the speed was increased to $250\\space m/min$, the time to complete the distance was reduced by $2$ minutes: $$200x=250(x-2)$$","Let's solve the resulting equation: $$200x=250x-500$$ $$50x=500$$ $$x=500:50=10$$<br />Then the length of the distance will be: $$200\\cdot 10=2000$$"]},{"content":"Hikers walk from their campsite to a rest stop at $6\\space km/h,$ and return at $4\\space km/h,$ The return trip takes $1 \\space h$ longer. How far is the rest stop from the campsite?[[image-64]][[fill_choice_big-757]]","widgets":{"image-64":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/11/turist-ryukzak-shlyapa-tsvetokelki-les.svg","width":"300"},"fill_choice_big-757":{"type":"fill_choice_big","options":["$12 \\space km$","$24 \\space km$","$36 \\space km$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the hiking time to the halt as $x.$<br />Then the tourists passed $6x\\space km.$","Returned to the camp site at a speed of $4\\space km/h,$, having spent $1 \\space h$ more on the return journey: $$6x=4(x+1)$$","$$6x=4x+4$$ $$2x=4$$ $$x=4:2=2$$ In $2$ hours tourists passed: $$6\\cdot 2 = 12$$"]},{"content":"A cyclist leaves point $A$. At the same time, a motorcyclist sets off behind the cyclist from point $B,$, which is $36\\space km,$ from point $A$. The cyclist travels at $6\\space km/h,$ and the motorcyclist at $15\\space km/h.$ At what distance from point $A$ will the motorcyclist catch up?[[image-49]][[fill_choice_big-798]]","widgets":{"image-49":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/03/na-velike-velosiped-01-2-3.svg","width":"299"},"fill_choice_big-798":{"type":"fill_choice_big","options":["$24\\space km$","$36 \\space km$","$12\\space km$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the time of movement of the cyclist as $x.$<br />Then the cyclist passed $6x\\space km.$<br />And the motorcyclist passed $15x\\space km.$","Before the meeting, the motorcyclist must travel an additional distance from point $B$ to point $A$:$$6x+30=15x$$","$$6x+36=15x$$ $$9x=36$$ $$x=36:9=4$$ The cyclist passed: $$6\\cdot4=24$$"]},{"content":"A truck leaves point $A$ at $45\\space km/h.$ After $3\\space h$, a car leaves point $A$ in the same direction at $60\\space km/h.$ At what distance from point $A$ will the car catch up with the truck?[[image-36]][[fill_choice_big-839]]","widgets":{"image-36":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/07/voditel-1.svg","width":"300"},"fill_choice_big-839":{"type":"fill_choice_big","options":["$540 \\space km$","$240 \\space km$","$320\\space km$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the time of movement of a passenger car as $x.$<br />The truck has therefore been travelling for $3$ more hours: $x+3.$<br />","The distance traveled by a passenger car is equal to the distance traveled by a truck: $$45(x+3)=60x$$","Let's solve the resulting equation: $$45x+135=60x$$ $$15x=135$$ $$x=135:15=9$$<br />We found the time of movement of a passenger car; it traveled in $9$ hours: $$60\\cdot9=540$$"]}]}

{"questions":[{"content":"Two bags each contain $100\\space kg$ of sugar. $4$ times as much sugar is taken from the first bag as from the second, leaving $60 \\space kg$ less sugar in the first bag. How much sugar remains in the second bag?[[image-1]][[fill_choice_big-511]]","widgets":{"image-1":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/10/sahar.svg","width":"300"},"fill_choice_big-511":{"type":"fill_choice_big","options":["$80 \\space kg$","$20\\space kg$","$50\\space kg$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take $x \\space kg$ sugar from the second bag.<br />Then from the first we took $4x \\space kg.$<br />","There is $60 \\space kg$ less sugar left in the first bag than in the second. Let's make an equation: $$100-4x+60=100-x$$","$$3x=60$$ $$x=60:3=20$$","After $20 \\space kg$ of sugar was taken from the second bag, it remained: $$100-20=80$$"]},{"content":"A student reads a $433$-page book in three days. On the second day, the student reads $25\\%$ more pages than on the first, and on the third day, $13$ more pages than on the second. How many pages does the student read on the first day?[[image-4]][[fill_choice_big-542]]","widgets":{"image-4":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/05/bloknot.svg","width":"299"},"fill_choice_big-542":{"type":"fill_choice_big","options":["$120$","$150$","$113$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the student read $x$ pages on the first day.<br />Then on the second day he read $1.25x$ pages.<br />On the third day he read $1.25+13$ pages.","The total number of pages in the book was $433$: $$x+1.25x+(1.25x+13)=433$$","Let's solve the resulting equation: $$x+1.25x+1.25x+13=433$$ $$3.5x=433-13$$ $$3.5x=420$$ $$x=420:3.5=120$$"]},{"content":"On the farm there are $700$ rabbits and chickens, they have $2000$ legs. How many rabbits are there on the farm?[[image-9]][[fill_choice_big-593]]","widgets":{"image-9":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/06/krolik.svg","width":"300"},"fill_choice_big-593":{"type":"fill_choice_big","options":["$300$","$200$","$100$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let there be $x$ rabbits on the farm.<br />Then chickens on the farm: $700-x.$","The animals have $2000$ legs in total: $$4x+2(700-x)=2000$$","Let's solve the resulting equation: $$4x+1400-2x=2000$$ $$2x=600$$ $$x=600:2=300$$"]},{"content":"The first plot has $4$ more raspberry bushes than the second. If $2$ bushes are moved from the second plot to the first, the first will have $2$ times as many bushes as the second. How many raspberry bushes are originally in the first plot?[[image-16]][[fill_choice_big-634]]","widgets":{"image-16":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/06/figurnyi-kust_ok-01.svg","width":"299"},"fill_choice_big-634":{"type":"fill_choice_big","options":["$14$","$15$","$16$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let there be $x$ raspberry bushes in the first area.<br />Then in the second area there are $x-4$ bushes.","Move $2$ bushes from the second plot to the first. The first plot then has twice as many bushes: $$x+2=2((x-4)-2)$$","Let's solve the resulting equation: $$x+2=2(x-6)$$ $$x+2=2x-12$$ $$x=14$$"]},{"content":"To return a library book on time, a student needs to read $30$ pages per day. The student reads $10 $ fewer pages per day and returns the book $6$ days late. How many days were originally available to read the book?[[image-100]][[fill_choice_big-675]]","widgets":{"image-100":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/02/student-uchenik.svg","width":"299"},"fill_choice_big-675":{"type":"fill_choice_big","options":["$12$","$11$","$10$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the number of days as $x. $<br />Then the student had to read a total of $30x$ pages.<br />","Instead, the student reads $20$ pages per day for $x+6$ days: $$30x=20(x+6)$$<br />","Let's solve the resulting equation: $$30x=20x+120$$ $$10x=120$$ $$x=120:10=12$$"]},{"content":"To complete an order on time, a factory needs to make $40$ items per day. It makes $20$ more items per day and finishes $3$ days early. How many days were originally allowed for the order?[[image-81]][[fill_choice_big-716]]","widgets":{"image-81":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/04/zavod-1.svg","width":"298"},"fill_choice_big-716":{"type":"fill_choice_big","options":["$9$","$8$","$7$"],"placeholder":0,"answer":0}},"step":1,"hints":["We will accept the order production time as $x.$<br />The factory needs to make $40x$ items in total.","By making $60$ items per day, it finishes $3$ days early: $$60(x-3)=40x$$","Let's solve the resulting equation: $$60x-180=40x$$ $$20x=180$$ $$x=180:20=9$$"]},{"content":"A number is increased by $6,$, the sum is multiplied by $2$, and $32,$ is subtracted from the product. The result equals the original number. What is the number?[[image-64]][[fill_choice_big-757]]","widgets":{"image-64":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/11/dumaet-primer-doska2-v2.svg","width":"300"},"fill_choice_big-757":{"type":"fill_choice_big","options":["$20$","$18$","$22$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the intended number as $x.$<br />Then after performing all these steps we should get the same number: $$x=2(x+6)-32$$","Solve the resulting equation: $$x=2x+12-32$$ $$x=20$$"]},{"content":"One tank contains $410\\space m^3$ of water and another contains $1\\space010\\space m^3.$ Every hour, $100 \\space m^3$ of water is added to the first tank and $20 \\space m^3.$ is pumped out of the second. After how many hours will the tanks contain equal amounts of water?[[image-49]][[fill_choice_big-798]]","widgets":{"image-49":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2024/04/urok_2.svg","width":"299"},"fill_choice_big-798":{"type":"fill_choice_big","options":["$5$","$4$","$12$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the number of hours as $x.$<br />Then after x hours there should be the same amount of water in the tanks: $$410+100x=1\\space010-20x$$","Solve the resulting equation: $$120x=600$$ $$x=600:120=5$$"]},{"content":"Students of the sixth and seventh grades made $315$ crafts for the exhibition. How many crafts did seventh graders make if it is known that they made $10\\%$ more crafts than sixth graders?[[image-36]][[fill_choice_big-839]]","widgets":{"image-36":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/09/bibloteka.svg","width":"300"},"fill_choice_big-839":{"type":"fill_choice_big","options":["$150$","$165$","$155$"],"placeholder":0,"answer":1}},"step":1,"hints":["Let's take the number of crafts made by sixth graders as $x.$<br />Then the seventh graders made $1.1x$ crafts.","Total number of crafts made: $315$: $$x+1.1x=315$$","Let's solve the resulting equation: $$2.1x=315$$ $$x=315:2.1$$ $$x=150$$","We found how many crafts sixth graders made. Now let’s find the number of seventh graders’ crafts: $$315-150=165$$"]},{"content":"A vehicle fleet initially has $2$ times as many trucks as cars. After $35$ cars are added and $10$ trucks are given to farmers, the fleet has $20$ more cars than trucks. How many vehicles were originally in the fleet?[[image-25]][[fill_choice_big-880]]","widgets":{"image-25":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/03/mashina.svg","width":"300"},"fill_choice_big-880":{"type":"fill_choice_big","options":["$75$","$50$","$60$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the number of passenger cars as $x.$<br />The number of trucks is $2x.$","After $35$ cars are added to the fleet and $10$ trucks are given to farmers, there are $20$ more cars than trucks: $$x+35=(2x-10)+20$$","Let's solve the resulting equation: $$x+35=2x-10+20$$ $$x=25$$ Initially, there were twice as many trucks: $$25\\cdot 2=50$$ Total number of vehicles in the fleet: $$25+50=75$$"]}]}

{"questions":[{"content":"A sweater, a hat, and a scarf used $550 \\space g$ of wool in total. The sweater used $5$ times as much wool as the hat, and the hat used $10\\space g$ more than the scarf. How much wool was used for the sweater?[[image-1]][[fill_choice_big-511]]","widgets":{"image-1":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/07/holodno-01-01.svg","width":"300"},"fill_choice_big-511":{"type":"fill_choice_big","options":["$400 \\space g$","$300\\space g$","$500\\space g$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let $x \\space g$ wool be used for the hat.<br />Then $5x\\space g$ wool was used for the sweater.<br />$x-10\\space g$ wool was used for the scarf.","We used $550\\space g$ wool for all things. Let's make an equation: $$x+5x+(x-10)=550$$","$$x+5x+x-10=550$$ $$7x-10=550$$ $$7x=560$$ $$x=560:7=80$$","The sweater used $5$ times as much wool as the hat: $$80\\cdot 5 = 400$$"]},{"content":"There are $166$ books on three shelves. The first shelf has $4$ fewer books than the second and $6$ more than the third. How many books are on the first shelf?[[image-4]][[fill_choice_big-542]]","widgets":{"image-4":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/04/flashcards.svg","width":"299"},"fill_choice_big-542":{"type":"fill_choice_big","options":["$56$","$60$","$76$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let there be $x$ books on the first shelf.<br />Then on the second shelf there are $(x+4)$ books.<br />And on the third shelf there are $(x-6)$ books.","In total there are $166$ books on the shelves: $$x+(x+4)+(x-6)=166$$","Let's solve the resulting equation: $$x+x+4+x-6=166$$ $$3x-2=166$$ $$3x=168$$ $$x=168:3=56$$"]},{"content":"$318$ cans of canned food are laid out in three boxes. In the third box there are $9$ more cans than in the first, and in the second there are $3$ fewer cans than in the third. How many cans of canned food are in the first box?[[image-9]][[fill_choice_big-593]]","widgets":{"image-9":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/10/shkaf-01.svg","width":"300"},"fill_choice_big-593":{"type":"fill_choice_big","options":["$101$","$200$","$110$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the first box contain $x$ cans.<br />Then in the third box there are $(x+9)$ cans.<br />The second box has three fewer cans than the third:$$(x+9)-3$$","In total there are $318$ cans in the boxes: $$x+(x+9)+((x+9)-3)=318$$","Let's solve the resulting equation: $$x+x+9+x+9-3=318$$ $$3x=318-15$$ $$3x=303$$ $$x=303:3=101$$"]},{"content":"The second garden plot has $5$ times as many currant bushes as the first. After $24$ bushes are moved from the second plot to the first, both plots have the same number of bushes. How many currant bushes were originally in the first plot?[[image-16]][[fill_choice_big-634]]","widgets":{"image-16":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/09/udobreniya-polivaet-rastenie-sad-ogorod-3.svg","width":"299"},"fill_choice_big-634":{"type":"fill_choice_big","options":["$12$","$15$","$11$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let there be $x$ bushes in the first section.<br />Then in the second area there are $5x$ bushes.","Move $24$ bushes from the second plot to the first. Both plots now have the same number of bushes: $$x+24=5x-24$$","Solve the resulting equation: $$4x=48$$ $$x=48:4=12$$"]},{"content":"A boat travels the same distance in $2 \\space h$ downstream as it does in $3\\space h$ upstream. Find the boat's speed in still water if the current flows at $3\\space km/h.$[[image-100]][[fill_choice_big-675]]","widgets":{"image-100":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/06/plavaniesudov2.svg","width":"299"},"fill_choice_big-675":{"type":"fill_choice_big","options":["$15 \\space km/h$","$20 \\space km/h$","$35 \\space km/h$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the boat's speed in still water be $x \\space km/h.$<br />The boat's downstream speed is then $x+3 \\space km/h.$<br />Its upstream speed is $x-3 \\space km/h.$","In $2$ hours with the current the ship covers the same distance as in $3$ hours against the current: $$2(x+3)=3(x-3)$$","Solve the resulting equation: $$2x+6=3x-9$$ $$x=15$$"]},{"content":"Two cars travel along a highway at the same speed. If the first increases its speed by $20 \\space km/h,$ and the second decreases its speed by $20 \\space km/h,$, the first will travel the same distance in $3\\space h$ as the second does in $6 \\space h.$ What is their original speed?[[image-81]][[fill_choice_big-716]]","widgets":{"image-81":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/10/gorod-lyudi-zanyatoj-mashiny-1-1.svg","width":"298"},"fill_choice_big-716":{"type":"fill_choice_big","options":["$60 \\space km/h$","$70\\space km/h$","$40\\space km/h$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the speed of the cars be $x \\space km/h.$<br />Then the speed of the first car will become $x+20 \\space km/h.$<br />And the speed of the second car will become $x-20 \\space km/h.$","In $3$ hours the first car will cover the same distance as the second in $6$ hours: $$3(x+20)=6(x-20)$$","Let's solve the resulting equation: $$3x+60=6x-120$$ $$3x=180$$ $$x=180:3=60$$"]},{"content":"A school repair team initially has $3$ times as many painters as carpenters. Four painters join the team and two carpenters leave to work elsewhere. The team then has $8$ times as many painters as carpenters. How many painters were on the team initially?[[image-64]][[fill_choice_big-757]]","widgets":{"image-64":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/10/rialiti-shou-01.svg","width":"300"},"fill_choice_big-757":{"type":"fill_choice_big","options":["$12$","$8$","$6$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let there initially be $x$ carpenters.<br />Then there were painters: $3x.$<br />After changing the crew, there were $3x+4$ painters, and $x-2.$ carpenters","There were $8$ times as many painters than carpenters: $$8(x-2)=3x+4$$","Let's solve the resulting equation: $$8x-16=3x+4$$ $$5x=20$$ $$x=20:5=4$$We found the number of carpenters. There were three times as many painters: $$4\\cdot 3 = 12$$"]},{"content":"There is a number written on the board. One student increased this number by $32,$ and the other decreased it by $2.$ The result of the first was $3$ times as large as the result of the second. What number is written on the board?[[image-49]][[fill_choice_big-798]]","widgets":{"image-49":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/04/uchitel-doska-globus.svg","width":"299"},"fill_choice_big-798":{"type":"fill_choice_big","options":["$19$","$17$","$45$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the required number as $x.$<br />One student increased this number by $32$: $x+32.$<br />The second student reduced this number by $2$: $x-2.$","It is known that one number turned out to be three times larger than another: $$3(x-2)=x+32$$","Let's solve the resulting equation: $$3x-6=x+32$$ $$2x=38$$ $$x=38:2=19$$"]},{"content":"There were $3$ $\\text{times}$ fewer grapes in the basket than in the box. After adding $1{,}5$ $kg$ grapes to the basket, the quantity in the basket became $2$ $\\text{times}$ less than in the box. How many kilograms of grapes were in the basket initially?[[image-36]][[fill_choice_big-839]]","widgets":{"image-36":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/10/obrazavr_frukty_vinograd.svg","width":"300"},"fill_choice_big-839":{"type":"fill_choice_big","options":["$3\\space kg$","$9\\space kg$","$2 \\space kg$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the basket initially contain $x$ $kg$ grapes. Then there were $3x$ $kg$ grapes in the box.","After $1{,}5$ $kg$ is added, the basket contains $(x+1{,}5)$ $kg$ of grapes. This is half the amount in the box: $$x+1{,}5=\\frac{3x}{2}.$$","Solve the equation: $$2x+3=3x$$ $$x=3.$$"]},{"content":"The first watermelon is $3 \\space kg$ lighter than the second. The third watermelon is $2$ times as heavy as the first. Together, the first and third watermelons are $1\\space kg$ heavier than the second. Find the mass of the first watermelon.[[image-25]][[fill_choice_big-880]]","widgets":{"image-25":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/08/arbuz-01-3.svg","width":"300"},"fill_choice_big-880":{"type":"fill_choice_big","options":["$2\\space kg$","$1\\space kg $","$3 \\space kg$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's take the mass of the first watermelon as $x.$<br />Then the mass of the second watermelon is $x+3.$<br />The mass of the third watermelon will be $2x.$","The first and third watermelons together are $1\\space kg$ heavier than the second: $$x+2x=(x+3)+1$$","Let's solve the resulting equation: $$3x=x+4$$ $$2x=4$$ $$x=4:2=2$$"]}]}

{"questions":[{"content":"A box contains $3$ times as many apples as a basket. After $5$ apples are moved from the basket to the box, the box contains $7$ times as many apples as the basket. How many apples were originally in the basket?[[image-1]][[fill_choice_big-511]]","widgets":{"image-1":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/10/obrazavr_frukty_yabloko.svg","width":"300"},"fill_choice_big-511":{"type":"fill_choice_big","options":["$10$","$7$","$15$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let there be $x$ apples in the basket, and $3x.$ in the box","Move $5$ apples from the basket to the box. The basket now contains $x-5$ apples, and the box contains $3x+5.$","The box now contains $7$ times as many apples as the basket. Write an equation:$$7(x-5)=3x+5$$","$$7(x-5)=3x+5$$ $$7x-35=3x+5$$ $$7x-3x=5+35$$ $$4x=40$$ $$x=40:4=10$$"]},{"content":"There are $56$ currant seedlings to divide among three teams. The second team should receive $2$ times as many seedlings as the first, and the third should receive $4$ fewer seedlings than the first. How many seedlings should the first team receive?[[image-4]][[fill_choice_big-542]]","widgets":{"image-4":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/09/udobrenie-udobryaet-rastenie-2.svg","width":"299"},"fill_choice_big-542":{"type":"fill_choice_big","options":["$15$","$25$","$14$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the first team be allocated $x$ seedlings. <br />The second team therefore receives twice as many seedlings: $2x.$ <br />And the third team was allocated $4$ fewer seedlings than the first: $x-4.$","There were $56$ seedlings in total: $$x+2x+(x-4)=56$$","Let's solve the resulting equation: $$x+2x+x-4=56$$ $$4x=56+4$$ $$4x=60$$ $$x=60:4=15$$"]},{"content":"One cinema box office sold $52$ more tickets than the other. How many tickets were sold at the first box office if a total of $212$ tickets were sold?[[image-9]][[fill_choice_big-593]]","widgets":{"image-9":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/05/kino-01.svg","width":"300"},"fill_choice_big-593":{"type":"fill_choice_big","options":["$132$","$85$","$80$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the first box office sell $x$ tickets.<br />Then at the second box office they sold $x-52$ tickets.","Total tickets sold: $212$: $$x+(x-52)=212$$","Let's solve the resulting equation: $$x+x-52=212$$ $$2x=212+52$$ $$2x=264$$ $$x=132$$"]},{"content":"A total of $17$ flood-damaged houses were restored on two streets. $3$ fewer houses were restored on the first street than on the second. How many houses were restored on the first street?[[image-16]][[fill_choice_big-634]]","widgets":{"image-16":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/03/idet-po-ulitse-vniz-1-2.svg","width":"299"},"fill_choice_big-634":{"type":"fill_choice_big","options":["$7$","$10$","$5$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let $x$ houses be restored on the first street.<br />Then $x+3$ houses were restored on the second street.","Total houses restored: $17$: $$x+(x+3)=17$$","Let's solve the resulting equation: $$x+x+3=17$$ $$2x+3=17$$ $$2x=17-3$$ $$2x=14$$ $$x=14:2=7$$"]},{"content":"A triangle has a perimeter of $18\\space cm.$ Two sides are equal, and each is $3 \\space cm$ longer than the third side. Find the length of each of the two longer sides.[[image-100]][[fill_choice_big-675]]","widgets":{"image-100":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/08/gora-01-1.svg","width":"299"},"fill_choice_big-675":{"type":"fill_choice_big","options":["$7\\space cm$","$4\\space cm$","$8\\space cm$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let each of the longer sides have length $x\\space cm.$<br />The shorter side then has length $x-3 \\space cm.$<br />","The perimeter of the triangle is $18\\space cm$: $$x+x+(x-3)=18$$","Let's solve the resulting equation: $$x+x+x-3=18$$ $$3x=18+3=21$$ $$x=21:3=7$$"]},{"content":"The length of the highway is $2\\space209\\space m.$. Most of the route is occupied by two tunnels, the length of one of which is $209\\space m$ greater than the length of the other. Find the length of the smaller tunnel if it is known that the surface part of the route is $500\\space m.$[[image-81]][[fill_choice_big-716]]","widgets":{"image-81":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/09/tonnel-mashina.svg","width":"298"},"fill_choice_big-716":{"type":"fill_choice_big","options":["$750\\space m$","$1\\space709\\space m$","$811\\space m$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the length of the smaller tunnel be $x\\space m.$<br />Then the length of the larger tunnel is $x+209\\space m.$<br />","The total length of the highway, including the section above ground, is $2\\space209\\space m$:$$x+(x+209)+500=2\\space209$$","Let's solve the resulting equation: $$x+x+209+500=2\\space209$$ $$2x=2\\space209-709$$ $$2x=1\\space500$$ $$x=1\\space500:2=750$$"]},{"content":"Four students bought bird food. The second student gave twice as much as the first, the third - three times as much as the second, the fourth - four times as much as the third, and all together gave $330$ rubles. How much money did the first student give?[[image-64]][[fill_choice_big-757]]","widgets":{"image-64":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/06/student-01-1.svg","width":"300"},"fill_choice_big-757":{"type":"fill_choice_big","options":["$10\\space \\text{rubles}.$","$15\\space \\text{rubles}.$","$20\\space \\text{rubles}.$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the first student give $x\\space \\text{rubles}.$<br />Then the second student gave $2x\\space \\text{rubles}.$<br />The third student contributed three times as much as the second: $2x\\cdot 3 \\space\\text{rubles}.$<br />The fourth student contributed four times as much as the third:$$(2x\\cdot 3)\\cdot4$$","All together gave $330\\space \\text{rubles}.$:$$x+2x+2x\\cdot3+2x\\cdot3\\cdot4=330$$","Let's solve the resulting equation: $$x+2x+6x+24x=330$$ $$33x=330$$ $$x=330:33=10$$"]},{"content":"Two workers made $99$ parts in total. The first worker made $20\\%$ more parts than the second. How many parts did the first worker make?[[image-49]][[fill_choice_big-798]]","widgets":{"image-49":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/09/shestirenka-2-01.svg","width":"299"},"fill_choice_big-798":{"type":"fill_choice_big","options":["$54$","$45$","$52$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the second worker produce $x$ parts.<br />Then the first worker produced: $(x+0.2x)$ parts.","Total $99$ parts produced: $$x+(x+0.2x)=99$$","Let's solve the resulting equation: $$x+(x+0.2x)=99$$ $$x+x+0.2x=99$$ $$2.2x=99$$ $$x=99:2.2=45$$","The first worker made $20\\%$ more parts: $$45\\cdot1.2=54$$"]},{"content":"The profit received by the company in the first two quarters of the current year amounted to $210 \\space000\\space \\text{rubles}.,$, and the profit received in the second quarter was $10\\%$ higher than in the first. How much profit did this company make in the first quarter?[[image-36]][[fill_choice_big-839]]","widgets":{"image-36":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/10/obrazavr_skuchayushhij.svg","width":"300"},"fill_choice_big-839":{"type":"fill_choice_big","options":["$100\\space 000\\space \\text{rubles}.$","$110\\space 000 \\space\\text{rubles}.$","$120\\space 000\\space \\text{rubles}.$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the profit received in the first quarter be $x\\space \\text{rubles}.$<br />Then in the second quarter the company received $(x+0.1x)\\space \\text{rubles}.$","Total profit amounted to $210 \\space000\\space \\text{rubles}.$: $$x+(x+0.1x) =210\\space 000$$","Let's solve the resulting equation: $$x+x+0.1x=210\\space 000$$ $$2.1x=210\\space 000$$ $$x=210\\space 000:2.1=100\\space 000$$"]},{"content":"Three schools received $87$ computers. The second school received $7$ more computers than the first, and the third received $13$ more computers than the second. How many computers did the first school receive?[[image-25]][[fill_choice_big-880]]","widgets":{"image-25":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/05/parta.svg","width":"300"},"fill_choice_big-880":{"type":"fill_choice_big","options":["$20$","$30$","$40$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let the first school receive $x$ computers.<br />Then the second one received $x+7$ computers.<br />The third school received $13$ more computers than the second: $x+7+13.$","Total schools received $87$ computers: $$x+(x+7)+(x+7+13)=87$$","Let's solve the resulting equation: $$x+x+7+x+7+13=87$$ $$3x+27=87$$ $$3x=87-27=60$$ $$x=60:3=20$$"]}]}